The $\oplus$3-REGULAR BIPARTITE PLANAR VERTEX COVER problem consists in computing the parity of the number of vertex covers of a 3-regular bipartite planar graph.
Which is the complexity of such problem? Is it $\oplus$P-hard, or is it in P?
What if we remove the planarity restriction (i.e. $\oplus$3-REGULAR BIPARTITE VERTEX COVER)?
The closest I was able to find is that $\oplus$3/2 BIPARTITE PLANAR VERTEX COVER is $\oplus$P-complete (see Theorem 2.2 in this paper).