Nonstandard dual parametrization of graph problems
I think this problem is FPT. Suppose that the graph contains a path on $2k+1$ vertices. Then, I claim the answer is YES: we select the second, fourth, sixth, etc. vertices of this path in a solution and remove them from the graph. We now have a graph $G'$ with $|E(G')|\le |E(G)|-2k$. It is easy to find a vertex cover of $G'$ with size at most $|E(G')|$. Together with the removed vertices this gives a vertex cover of size at most $|E(G)|-k$ for $G$.

If the graph does not contain a path on $2k+1$ vertices, then a DFS tree of the graph has height at most $2k+1$, and hence can be used to construct a tree decomposition of width at most $2k$. We can then solve Vertex Cover optimally with the standard algorithm for treewidth.