$\oplus$SAT is the problem of deciding if the number of satisfying assignments to a CNF formula is odd (and is the standard complete problem for the class $\oplus$P, or Parity-P).
Suppose we have a fast algorithm for SAT. Is there anything known about what this algorithm would imply for $\oplus$SAT?
More concretely, suppose that P=NP. Does this imply that $\oplus$P collapses to some (presumably) smaller complexity class?