Is there an oracle separating Parity-P from PSPACE?
From the comment above: you can use the same oracle $A$ used in:

Richard Beigel, Harry Buhrman, and Lance Fortnow. 1998. NP might not be as easy as detecting unique solutions. In Proceedings of the thirtieth annual ACM symposium on Theory of computing (STOC '98).

for which $P^A= \oplus P^A  \neq NP^A = EXP^A$  ($EXP^A=NP^A\subseteq PSPACE^A$ also holds)