Enumerating all pairs of disjoint paths
I believe Artem Kaznatcheev's answer is correct, but it does not give polynomial space. So here's a different approach that should work a little better in that respect.

Using max flow it is possible to solve a slightly more general problem: find a pair of edge disjoint paths from some two vertices {s1,s2} to some other pair of vertices {t1,t2}, but without controlling which source vertex is connected to which destination vertex.

Suppose we have a graph G and vertices s1,s2,t1,t2 for which we want to list all pairs of paths. Find a single pair of paths P1,P2, and let e = (s1,v) be the first edge on one of those paths. Then we can split the problem space into two subproblems: the pairs of paths that use e are the same as the paths from {v,s2} to {t1,t2} in G-s1, and the pairs of paths that do not use e are the same as the paths from {s1,s2} to {t1,t2} in G-e. Recurse in both of these two subproblems, and (to avoid duplication) only report a path when you're at the bottom of the recursion.