In this paper we give efficient parallel algorithms for solving a number of visibility and shortest path problems for simple polygons. Our algorithms all run in Ο(log n) time and are based on the use of a new data structure for implicitly representing all shortest paths in a simple polygon P, which we call the stratified decomposition tree. We use this approach to derive efficient parallel methods for computing the visibility of P from an edge, constructing the visibility graph of the vertices of P (using an output-sensitive number of processors), constructing the shortest path tree from a vertex of P, and determining all-farthest neighbors for the vertices in P. The computational model we use is the CREW PRAM.