Summary: We present a parallel randomized algorithm running on a CRCW PRAM, to determine whether two planar graphs are isomorphic, and if so to find the isomorphism. We assume that we have a tree of separators for each planar graph (which can be computed by known algorithms in \(O(\log^2 n)\) time with \(n^{1+\varepsilon}\) processors, for any \(\varepsilon> 0\)). If \(n\) is the number of vertices, our algorithm takes \(O(\log(n))\) time with \(P= O(n^{1.5}\cdot \sqrt{\log(n)})\) processors and with a probability of failure of \(1/n\) at most. The algorithm needs \(2\cdot\log(m)- \log(n)+ O(\log(n))\) random bits. The number of random bits can be decreased to \(O(\log(n))\) by increasing the number of processors to \(n^{3/2+\varepsilon}\), for any \(\varepsilon> 0\). Our parallel algorithm has significantly improved processor efficiency, compared to the previous logarithmic time parallel algorithm of \textit{G. L. Miller} and \textit{J. H. Reif} [SIAM J. Comput. 20, No. 6, 1128-1147 (1991; Zbl 0737.68066)], which requires \(n^4\) randomized processors or \(n^5\) deterministic processors.