Very much this. When in doubt, look at what the PHB says under “using each ability.” On the subject of Strength, it says:Keep in mind that 5E presents checks as ability checks, not skill checks. So what you're actually supposed to be doing is saying "What ability works here? Strength or Dexterity?" then ask "Is a character Proficient in X and would it apply?"
This allows you figure out what any character can do, and the those that are proficient are better at that thing. Works for any skill.
And on the subject of Dexterity, it says:
It’s also worth noting that in 5e, characters can climb at half their speed and jump a specified distance based on their Strength score without need for a roll, provided there aren’t complicating factors that would make the outcome uncertain and risky. So, the majority of parkour activities probably don’t need a roll. When a particular parkour-related action does have an uncertain outcome and consequences for failure, which ability it is most appropriate to call for a check with will depend on the specific circumstances. If they need to keep balance or move quickly or nimbly to succeed, Dexterity. If they need to apply a lot of direct physical force to succeed, Strength. In either case, if the character is Proficient in Athletics, they can add their proficiency bonus if the task involves climbing, jumping, or swimming. If they’re proficient in Acrobatics, they can add their proficiency bonus if the task involves trying to stay on their feet, or to dive, roll, somersault, or flip.
Personally, I think the majority of parkour-related actions that require a roll to resolve, e.g. Prince of Persia style wall running between the moving saw blades would be Dexterity (moving quickly/agilely) plus Athletics (Running/climbing/jumping). But it will always depend on the particulars of the action. What’s the source of danger, what is the character doing to try to overcome it, why is the outcome uncertain, and what do they need to perform well at to be successful in the face of that uncertainty?