Best known joint containments for/by 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]