$\mathsf{EXP}$ vs $\oplus\mathsf{EXP}$
In terms of complexity reasons (rather than complete problems): The Hartmanis-Immerman-Sewelson Theorem should also work in this context, namely: $\mathsf{EXP} \neq \oplus \mathsf{EXP}$ iff there is a polynomially sparse set in $\oplus \mathsf{P} \backslash \mathsf{P}$. Given how far apart we think $\mathsf{P}$ and $\oplus \mathsf{P}$ are - e.g. Toda showed that $\mathsf{PH} \subseteq \mathsf{BPP}^{\oplus \mathsf{P}}$ - it would be quite surprising if there were no sparse sets in their difference. 

More directly, if there were no sparse sets in their difference, it would say that for every $\mathsf{NP}$ verifier, if the number of strings of length $n$ with an odd number of witnesses is bounded by $n^{O(1)}$, then the problem [of telling whether there is an odd number of witnesses] must be in $\mathsf{P}$. This seems like quite a striking and unlikely fact.