Parity-P is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths (rather than a zero or non-zero number of acceptance paths). Thus Parity-P is basically PP's stunted younger sibling: while PP counts whether or not the number of accepting paths of an NP-machine is a majority or not (i.e. the most-significant bit of that quantity), Parity-P indicates the least-significant bit of the number of accepting paths.
Like NP, Parity-P contains UP (which contains P, "probably" strictly so); and like NP, Parity-P is contained in PSPACE.
Question. What are the best known joint upper and lower bounds for NP and Parity-P?