Complexity of parity game solving compared to PLS, PPA, and PPAD
Yes, solving parity games is known to be in PPAD (and thus PPA and PPP too) and PLS, and is thus unlikely to be hard for either (since this would imply containment of one of these classes in the other).

See, e.g., 

Daskalakis, Constantinos, and Christos Papadimitriou. "Continuous local search." Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms. SIAM, 2011.

and combine the membership of Simple Stochastic Games (SSGs) in CLS (which is in PPAD and PLS) with the well-known observation that solving parity games can be reduced to solving SSGs in polynomial time.

The reason that these problems are in PPAD is that they admit "optimality equations", rather like Bellman equations, that characterize solutions as fixed points. The reason these problems are in PLS is that they can be solved with local improvement algorithms like strategy improvement (a two-player generalization of policy iteration for MDPs).