Best known joint containments for/by NP and Parity-P? 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?
By Valiant-Vazirani, NP is contained in BP dot Parity-P (which obviously contains Parity-P).
Moreover, Toda showed that PH is in BP dot Parity-P which is in P^(#P) (which is in PSPACE).

For lower bounds, I think both classes contain a class known as FewP which contains UP and is like NP but you ask that strings in the language have at most polynomially many accepting paths.

[Update: corrected typo BPP instead of BP]