An outerplanar graph is a planar graph that can be imbedded in the plane in such a way that all vertices lie on the exterior face. An outerplanar graph is maximal if no edge can be added to the graph without violating the outer-planarity. In this paper, an optimal parallel algorithm is proposed on the EREW PRAM for testing isomorphism of two maximal outerplanar graphs. The proposed algorithm takes O(log n) time using O(n) work. Besides being optimal, it is very simple. Moreover, it can be implemented optimally on the CRCW PRAM in O(1) time.