Parallel processing has been applied for time domain simulations of power system transient behavior in order to implement real-time dynamic security assessment. In this paper, two different algorithms have been implemented and compared: the Shifted-Picard (SP) and the Very DisHonest Newton (VDHN). The former has been proved to be effective when parallelism-in-time is adopted whereas the latter is an effective solver when parallelism-in-space is exploited. Furthermore, two different parallel computing architectures have been considered: namely, the Sequent Symmetry computer with 26 processors which is a data shared memory machine and the nCUBE characterized by 128 CPUs which is a typical message passing parallel machine. A realistic network with 662 buses has been used to assess the performance of the different implementations. The comparison of the results allows the reader to understand both the limitations of the algorithmic approaches and the constraints imposed by the two parallel architectures. An optimal grain of the parallelism associated to the problem can be identified through the reported experience.