As the need for faster power system dynamic simulations increases, it is essential to develop new algorithms that exploit parallel computing to accelerate those simulations. This paper proposes a parallel algorithm based on a two-level, Schur-complement-based, domain decomposition method. The two-level partitioning provides high parallelization potential (coarse- and fine-grained). In addition, due to the Schur-complement approach used to update the sub-domain interface variables, the algorithm exhibits high global convergence rate. Finally, it provides significant numerical and computational acceleration. The algorithm is implemented using the shared-memory parallel programming model, targeting inexpensive multi-core machines. Its performance is reported on a real system as well as on a large test system combining transmission and distribution networks.