Sylow subgroups are fundamental in the design of asymptotically efficient group-theoretic algorithms, just as they have been in the study of the structure of finite groups. We present efficient parallel (NC) algorithms for finding and conjugating Sylow subgroups of permutation groups, as well as for related problems. Polynomial-time solutions to these problems were obtained more than a dozen years ago, exploiting a well-developed polynomial-time library. We replace some of those highly sequential procedures with ones that work through a polylog-length normal series that is a by-product of NC membership-testing. As in previous investigations, we reduce to the base case of simple groups, and handle this by a case-by-case analysis that depends heavily upon the classification of finite simple groups.