Parahuman Powers, the Atomic Robo Way
You're right in that the core rules gain a huge benefit from being a general framework to drop stunts on. Stunts are a way to express your individuality, and if they get too involved you can find yourself bogged down in the minutiae of stunt wrangling, the question of what applies, and when, and how, instead of focusing on what the character's doing.
However, there are a couple gaps in there that I feel Atomic Robo patches up pretty well. It's a pulp-adventure game with a science bent by default, but it's sufficiently setting-agnostic that I tweaked a couple of skills and set it down to power a fantasy game with nary a hiccup. So what's it do?
Bonus Stunt Vocabulary
This is going to be important later.
Your Signature Aspect is a stunt you drop to highlight one of your aspects. This can be your high concept, but it doesn't have to be. You get one free invoke every session on the aspect, but more importantly, when the GM compels it, they have to start their offer at two fate points. This interacts very interestingly with some later bits.
A Gear Stunt represents a unique piece of gear you have guaranteed access to when it would be reasonable. It can roll up multiple stunt slots into one giant benefit - every stunt spent on gear gives you two mini-bennies: Weapon:2 (limit 4), Armor:1 (limit 2), a +1 bonus respecting the normal stunt scope rules for +2s, or a skill swap with a narrower scope than one stunt. The game itself's got plenty of worked examples of these in the rules proper.
But let's talk about the big deal:
Specifically as they relate to aspect permissions.
We need to start at the central conceit of Fate, which is that the numbers are completely made up. They're attached to, at best, adjectives. You can start a game of Fate as a mouse or as a cosmic consciousness, and as long as all your fellow players are the same and you're facing troubles that a mouse or a cosmic consciousness, respectively, would be able to solve, nothing about the numbers has to change.
You can think of the difficulty of any conceivable action as lying along a particular spectrum, and it's divided up into three buckets:
$$\textit{you can't} \lt \textit{let's roll} \lt \textit{you can}$$
Can and can't stretch off into infinity, but the little bucket of "let's roll" in the middle is where the outcome is in doubt. Aspect permissions deal with where that bucket winds up.
The problem with trying to meaningfully discuss aspect permissions is that most games of Fate put everybody at the same general level of competence. Fantasy adventurers can all handle themselves in a fight, investigators can all talk to witnesses and look for clues, and any given member of the Superfriends can get through anything the writers want them to:
Last Son of Krypton lets Superman Forcefully punch through a steel wall, and Partially Fused With the Speed Force lets the Flash Quickly phase through it.
The function of aspect permission in games that deal largely among equals is to rattle the bucket around a bit - something just a little too hard can now be rolled for, something a little too easy is just straight-up done.
Zird The Arcane just recognizes the major signifiers (dress, speech, comportment) of the notable world cultures without needing to roll an easy Lore check; for Landon, who was trained by the White Order, this is true as pertains to the common weapons of the world and their signature fighting styles.
Likewise, Zird can roll Lore to read the motives of someone presenting a culturally-significant gift, and Landon can roll it to realize someone is lying about who they are if their weapon would be a giveaway.
Hey, wait a second. Isn't there something else that works kind of like the opposite of this? It takes a situation where you would be getting on just fine and suddenly you've got some dice to roll, or one where the dice would give you a chance, but suddenly it just doesn't work out for you, darn the luck? That's a compel, right? It gets you a fate point, right?
Guess what you're doing when you're spending a fate point to declare a story detail. They're both shifts of that "let's roll" bucket, just in opposite directions. Generally speaking, unless something was right on the edge one way or the other, neither a compel nor a story detail is going to jump you direct from "you can't" to "you can", or vice-versa.
Atomic Robo's aspect permission extra is called the Weird Mode. Characters are built using three skill modes at +3, +2, and +1, and there are four stock modes that it's assumed most action scientists up for banter and intrigue are going to be equally capable in. But you can be something more than that. You can be a Special Agent, or a Celebrity, or a Reporter... or a Robot, Shark, or Dinosaur. Each of your modes gives you one of your character aspects, and any aspect attached to a Weird Mode can be a pretty significant shift of the bucket, usually as it relates to some skill that's in the Weird Mode.
Normal people would need to roll Physique to function underwater for a dramatic period of time, but Special Agent Robo doesn't need to breathe and can keep trundling on without a care.
Normal people can't punch through a steel wall, but Special Agent Robo can play crack_knuckles.wav and roll Physique to try and Overcome it.
Stunts and Limitations
What is a Stunt but a Fate Point you're always spending, in some particular circumstance? All the +2 Stunts should make that pretty obvious, but the skill swap stunts should also make sense in that regard now that I've talked about using a Fate Point to declare a story detail and shift the bucket. Fate Accelerated makes it explicit that all of your stunts are tied to some character aspect, and that requirement isn't necessarily there for Fate Core, but it's hard to imagine taking a stunt that didn't relate to anything about the character. Know what else is in Fate Accelerated? Stunts that look like this:
Here All Along: Because I am a Master of Disguise, once per session, in a scene I am not currently in, I can declare that I am there in disguise as a minor character otherwise beneath notice, if it would be remotely possible.
Signature Aspect also gives you something once per session: a free invoke. Yep, this is a Stunt that also needs a Fate Point, because it's making two jumps. Normal people can't get into somewhere disguised as a faceless mook. A Master of Disguise would be able to spend a point and roll for it, which is the stunt, and the 1/session virtual Fate Point take it one more step, to where they just can, even without rolling.
It's pretty hard to gauge a jump of even two steps except in this way, from can't to can. If you have the idea that a chain of stunts might be possible, like one stunt gives you fluency in all known languages and one after that lets you talk to anything that has a language, even aliens, spending Fate Points to jump one stunt ahead to where you're rolling to understand the display screens in a strange metalloid protrusion from the barrier peaks isn't a bad use of declaring a story detail. But "from can't to can" is the general form a lot of parahuman powers are going to wind up taking. So! How many times a session is this power going to be useful, do you think?
1-3 times? Then a stunt's fine, either a "once a session" stunt or one that lets you spend a Fate Point and the job's done. Once a scene or even more? Atomic Robo steps in there with Weird Skills and Mega-Stunts, depending on whether what you're doing is still in the general power neighborhood but tangential to human normalcy or generally a cut above. Not even once a session? Robo's there, too, with its schemes for invention and procurement.
Weird Skills And Mega-Stunts
Atomic Robo used a point-buy system to tack some extra skill bumps on the end of its skill mode creation; skills were costed based on how broadly they were applicable. Things that the Weird Modes were capable of that didn't really have ready analogies to human normal skills were made into Weird Skills; Dinosaurs and Sharks weren't capable of the full range of human Combat, but they do have Tooth and Claw (or perhaps Tooth and Tail) which did about the same things. Spirits could Hover, which was kind of a fusion of Athletics and Stealth, doing some of each. If your power is something abnormal for humans but something any of your para-kind can do, it might be worth modeling as a weird skill.
A mega-stunt is tied to one of your Aspects - you can't have Atomic Strength without being Special Agent Robo. Megas can be as strong as multiple regular stunts stuck together, and they still all count as one stunt, but for every point over, the GM starts each session with an extra reserve point they can dump into scene opposition at any time. (The reserve's another invention that also gives the GM a formal place to hold all those FP their characters get from conceding fights or eating compels.)
Megas can replicate any other stunt effect except for Signature Aspect. When used for weapons and armor they give you Weapon:2 and Armor:2, same limits as gear. But they can also go beyond.
Break The Limit. Megas will let you hit Weapon:6 and Armor:4 if you want.
Absolute Defense. Megas can render you immune to a particular type of offense. When someone comes at you with guns, or fire, or lies, or threats, you never take Stress from them. (The attack still happens and your attacker might get a boost as a result.)
Absolute Capability. Megas can let you impose your will almost unilaterally. An opposed overcome using your prodigious strength, peerless insight, or uncanny stealth will always be treated as a normal success, unless you're up against another Absolute.
However, this is always "at a cost". Megas take you from can't to can, not at the cost of a Fate Point, but at the cost of a Fate Point's worth of setback. You're asking the GM for a compel on the aspect associated with them - you see how Signature Aspect interacts with this. Defenses and armor limits breaks may alternately come with an unusual weakness - a vulnerability to something (like electromagnetic fields) or a difficulty in something (like working with inanimate objects) that normal people don't have. This can be a plot compel, but it can also skirt the edge of one - not a Fate Point's worth, but still a reason for opposition to exist for you and not for someone else. So, for instance:
Atomic Strength. Absolutely stronger than any normal human, but at a cost.
I Can Hear Your Heart. No one can lie to you, but you have difficulty perceiving anything without a heartbeat.
But in this world of Action Science, even regular people can get access to these para-powers. How?
MEGAWEAPON
Gear can also be built mega-grade instead of the normal sort. The Aspect the stunts are linked to is not any aspect on the character, but rather something in the function or flaw of the gear - basically its own high concept and trouble.
Experimental Plasma Cannon. Function: Destructive Lightning Blaster. Flaw: Limited Battery Pack. Weapon:4, or Weapon: 6 but at a cost.
Boots of the Zephyr. Function: Run like the Wind. Flaw: Lighter than Air. Absolutely faster than any normal human, but at a cost. +2 to use Athletics to Overcome and Create an Advantage through swift running and high or long jumping.
(both of these elements of Gear are built on 3 stunts, so if PCs had these stunts the GM would start the session with 2 bonus points in the reserve pool for each one.)
Mega-scale gear can also open up access to weird skills through skill swaps:
SPECTRE Suit. Function: Ectoplastic Bodysuit. Flaw: Beacon to the Spirit World. +2 to Stealth attempts to Defend against detection through sight or sound. You can use Stealth to Overcome or Create an Advantage by partially or fully phasing through barriers, as with Hover.
Rocket Pack. Function: Rocket Packs Are Easy. Flaw: Calf Shields Are Hard. +2 to attempts to Overcome horizontal or vertical gaps with Athletics. You can use Vehicles to Overcome or Create an Advantage by straight-up flying or precision maneuvering, as with Hover.
Temporary Stunts with Invention and Procurement
Robo created mechanics for scientists to invent gear as the travails of Action Science demanded it, and in the Majestic-12 expansion, let agents of an intelligence organization "requisition gear", sometimes not even with sarcasm quotes. These are mechanics, with rolls to make and costs to pay, but one of the costs, for getting what was basically an extra stunt or more when you needed them, was to fill up the GM reserve proportional to how many stunts were in the thing you were making or getting. If you needed the gear over multiple sessions, the reserve filled up every time.
So if your parahuman power is really a bunch of little situational ones, you might consider just saying that you're able to create them as needed, with at minimum some compensation going into a GM's reserve and maybe an additional mechanic that makes sense gating access.
