The subgraph homeomorphism problem for a fixed graph $H$ is stated as follows: given a graph $G$, determine whether $G$ has a subgraph homeomorphic to $H$, and obtain it. We study the parallel complexity of this problem for various pattern graphs $H$ and present fast $NC$ algorithms for versions of this problem. We also present an efficient $NC$ algorithm to check if a given graph is outer-planar and to obtain its forbidden homeomorphs $K_{4}$ or $K_{2,3}$ if it is not.