We give an $NC$ algorithm for finding vertex disjoint $s_{1}, t_{1}$ and $s_{2}, t_{2}$ paths in an undirected graph $G$. An important step in solving the general problem is solving the planar case. A new structural property yields the parallelization, as well as a simpler linear time sequential algorithm for this case. We extend the algorithm to the non-planar case by giving an $NC$ algorithm for finding a Kuratowski homeomorph, and, in particular, a homeomorph of $K_{3}, 3$, in a non-planar graph. Our algorithms are processor efficient; in each case, the processor-time product of our algorithms is within a polylogarithmic factor of the best known sequential algorithm.