In this paper, a simple parallel algorithm for finding all the maximal cliques of an interval graph is proposed. This algorithm can be implemented in parallel in O(log n) time using O(n2/log n) processors. The maximal cliques of an interval graph contain important structural information. Many problems on interval graphs can be solved after all the maximal cliques are known. It is shown that cut vertices, bridges, vertex connectivities, and vertex-disjoint paths can all be determined easily after the maximal cliques are known. Finally, the all-pair shortest path problem for interval graphs is solved based on the relationship between maximal cliques. The all-pair shortest path algorithm can also be parallelized in O(log n) time using O(n2/log n) processors.