Simulating a power system with both transmission and distribution networks modeled in detail is a huge computational challenge. In this paper, we propose a Schur-complement-based domain decomposition algorithm to provide accurate, detailed dynamic simulations of the combined system. The simulation procedure is accelerated with the use of parallel programming techniques, taking advantage of the parallelization opportunities inherent in domain decomposition algorithms. The proposed algorithm is general, portable and scalable on inexpensive, shared-memory, multi-core machines. A large-scale test system is used for its performance evaluation.