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