How hard is to compute $\Delta_{|V|}$?
As the OP pointed out in an update above, the second question about computing $\Delta_{|V|}$ is answered in this question.

The first question is about computing the parity of the number of vertex covers. This problem is $\bigoplus$P-hard even for planar 3-regular graphs.  See Section 4.1 of this paper (particularly Corollary 4.2).

First, the notation for the corresponding Holant problem is Pl-$\bigoplus$Holant$([0,1,1]|[1,0,0,1])$.  Since there is a 1-to-1 correspondence between vertex covers and independent sets, sometimes people also refer to the Holant expression for independent set (in this case, $\bigoplus$Holant$([1,1,0]|[1,0,0,1])$) as though it were the vertex cover problem.

Second, I don't know why they called this problem $\bigoplus$3/2-Bip-VC (that is, why "3/2-Bip").  As I said above, this Holant expression is for the counting of vertex covers (equivalently, independent sets) in 3-regular graphs.