The Pinned condition specifically says you are very limited on your actions:
A pinned creature is tightly bound and can take few actions.
A pinned creature is limited in the actions that it can take.
So unless something says otherwise, you are limited to the options on the condition, which are listed right after that:
- A pinned creature can always attempt to free itself;
- A pinned creature can take verbal and mental actions, but cannot cast any spells that require a somatic or material component.
So you could take Swift Actions if those are verbal or mental or attempt to free yourself. But since it does not say "You can take swift actions", you are limited to only those swift actios that do specify that they are purely mental or verbal in nature, or have a description that suggests that the character only has to speak or think for it to work, such as:
Every other swift action should be impossible, such as an alchemist trying to apply poison to a weapon, or the same warpriest enhancing his weapon with magical power.
Lay on Hands specifically says that you need a free hand, which by the definition of the Pinned condition, you probably won't have (but you would while grappled).
Despite the name of this ability, a paladin only needs one free hand to use this ability
Regardless, the character still should be allowed to use a spell-like ability (no somatic or material components) if they pass their concentration check.
This seems to be the concensus of the community aswell, as can be observed here, here, and finally here. On the last, Sean K. Reynolds state that a touch attack is not a mental or verbal action and the thread's FAQ request was answered "Staff response: no reply required.".
James Jacobs, Creative Director at paizo already stated that you can't even attack while pinned:
Can a pinned creature or character make a full attack?
Nope, because one of the things that happened when you are pinned is that you are VERY limited as to what actions you can take. Most importantly, you can't attack if you're pinned—in order to attack, you first need to escape the pinned condition. So... no. If you're pinned, you can't make a full attack.
For all purposes, the rules as intended is that you can only take the actions listed on the Pinned condition.
If the intent of Pinned was to allow you to take the same actions allowed by grapple, the devs wouldn`t even mention that you can take mental and verbal actions, as those are already allowed while grappled. They would simply state that you can only cast spells without somatic components and take a heavier penalty to AC.
This would save text (like half of it) and still be clear about the intent of the condition.