My point of view is that of a professional persuader. It's what I do for a living. A lot of other people do as well, and they may have different views on this. I'd like to suggest two things, though, that would make my approach more "correct." One, I'm very introspective so I've included to the best of my ability tacit learning from experience that is difficult to codify. And I'd also suggest that it is possible a successful persuasion may appear on the surface to be more similar to a different approach, but it also fits my approach, and when you concern yourself at the goings on beneath the surface, I believe this system most accurately describes how persuasion is actually done.
I break down the anatomy of persuasion into 5 phases. Introduction, Discovery, Presentation, Elaboration, and the Close. I'll get to the skills in a bit. If you understand the process, the skill list almost flows naturally from it.
Introduction - Need not be a literal introduction. You might be persuading someone you already know. The idea at this stage is to get their attention and willingness to listen.
Discovery - In classical persuasion, you discover what the other person wants by talking to them. Usually it involves direct questioning. But it could also be more subtle. Getting a "read" on the other person during an interaction. Of course, in some cases, discovery can happen outside of the person's presence. Investigation and any number of normal adventuring skills may apply to gathering the necessary information.
Presentation - This is the phase that most people think of as being the beginning, middle, and end of persuasion. If you're not always very persuasive in real life, it's because you skip right to this phase. Without knowing for sure what the other person wants, it is possible to hit on the right things by dumb luck. But that's all it is. Luck. Not skill. The skill involved in the presentation phase is about clearly communicating the proposition and framing it in a way that is appealing.
Elaboration - This phase is optional, and will be initiated by the person you're trying to persuade. They're not 100% convinced, so they still have some questions. The nature of the questions may signal to the persuader that he failed to collect all the necessary information in the discovery phase, and a skilled persuader must fall back to that to avoid failure.
Close - Typically a simple yes or no question. If you've done everything right up to this point, it should be an easy yes. If it's a no, the skilled persuader must once again fall back to the discovery phase to avoid failure.
With this in mind, the vital skill areas are:
1) Haling/Parleying/Charisma/Oration to gain that initial opportunity,
2a) Cold Reading/Lie Detection OR 2b) Questioning/Interviewing/Interrogation depending upon how you are gathering your information,
3a) Parleying/Charisma/Oration OR 3b) Lying/Con/Deception depending on whether or not what you're proposing actually does fit the other person's interests,
4a) Cold Reading/Lie Detection OR 4b) Parleying/Charisma/Oration OR 4c) Lying/Con/Deception depending on whether you're answering the questions truthfully or with lies, or whether the person you're trying to persuade is being truthful and sincere with their questions and objections.
5) Parleying/Charisma, but only if the persuasion attempt is still in doubt, otherwise success is automatic without further skill check.
So to boil the above down to a short list, I would use:
1) Parleying - Covers engaging, effective, and purposeful communication
2) Detect Lie - Covers not just lie detection but also sensing motives
3) Interview - Covers interrogation and effective questioning skills to gain desired information.
4) Deception - Covers knowingly passing false information with conviction
Each of these skills are equally important. I think it's natural to assume Parley would be the alpha social skill under this scheme, but it is not so. Remember, without knowing what it is the other person wants, persuasion comes down to dumb luck, skill is irrelevant. This makes parleying only useful in the introduction, to get a chance to "pitch" so to speak if the person isn't skilled at interview.
Detect lie is also important for two reasons. One, not everyone can be persuaded in a particular situation. Part of being a skilled persuader is finding out sooner rather than later than you're wasting your time. Two, it indicates that the all-important discovery phase may need to be revisited. Three, the biggest pit-fall in persuasion is not getting a "No" but rather the person agreeing just to shut you up then not following through with what they agreed to. A persuader worth his salt needs to be able to pick up on that.
Finally, deception, while completely unnecessary if you choose to persuade strictly honestly, can be the only means of convincing someone when it is not possible to find a proposition that serves their best interests as well as yours. It can also be used as a crutch in cases when perhaps there is a way for mutual gain but the persuader just can't think of what that might be at the moment.