We present a cost-optimal parallel algorithm for the maximum matching problem on bipartite permutation graphs on an EREW PRAM. Previously, Chen and Yesha have dealt with this problem. Their solution relies on Dekel and Sahni's matching algorithm for convex bipartite graphs, which runs in O(log2n) time usingO(n) processors. Given a permutation diagram, our algorithm runs in O(logn) time by using O(n/logn) processors. Our method starts with an easily understood greedy algorithm. We define a nontrivial binary operation which is associative and equivalent to the greedy algorithm. Thus parallel prefix can be applied to the problem.