Nonblocking multistage interconnection networks are favored to be used as switching networks whenever possible. Crosstalk-free requirement in photonic networks adds a new dimension of constraints for nonblockingness. Routing algorithms play a fundamental role in nonblocking networks, and any algorithm that requires more than linear time would be considered too slow for real-time applications. One remedy is to use multiple processors to route connections in parallel. In this paper, we study the connection capacity of a class of rearrangeable nonblocking and strictly nonblocking networks with/without crosstalk-free constraint, model their routing problems as weak or strong edge colorings of bipartite graphs, and propose efficient routing algorithms for these networks using parallel processing techniques.