Complexity of $\oplus$ 3-REGULAR BIPARTITE PLANAR VERTEX COVER 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.


  Question
  
  
  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).
Ok, according to the paper Accidental algorithms by Valiant, the  problem ⊕Pl-3/2Bip-Mon-2CNF is ⊕P-complete. So ⊕ 3-REGULAR PLANAR VERTEX COVER would also have to be ⊕P-complete. So it's not in P (unless P = ⊕P-complete).
But I still don't know if ⊕ 3-REGULAR BIPARTITE PLANAR VERTEX COVER is ⊕P-complete.