Parallel algorithms for reachability in directed planar graphs
See


Kao, Ming-Yang; Klein, Philip N. (1993), Towards overcoming the transitive-closure bottleneck: efficient parallel algorithms for planar digraphs, J. Comput. System Sci. 47 (1993), no. 3, 459–500.


Their Theorem 10 gives a deterministic CRCW algorithm for $st$-reachability in planar digraphs with $O(n)$ processors and $O(\log^3 n)$ time. Searching Google scholar for other papers that cited theirs, I didn't see anyone else with improvements to this.