This paper presents a decoupled space-time parallel solver that integrates the Parareal algorithm (parallelin-time) with the Multi-Area Thevenin Equivalent (MATE) method (parallel-in-space) for simulating the transient stability of large-scale power systems. A shared-memory implementation of MATE, exploiting both spatial and task-level parallelism, is incorporated into the Master-Worker (PM) and Distributed (PD) Parareal paradigms. The hybrid solver concurrently employs two high-performance computing (HPC) frameworks-OpenMP for MATE and message passing interface (MPI) for Parareal-to achieve scalability across both domains. Two implementation strategies are examined: homogeneous configurations with equal MATE partitions and heterogeneous configurations with unequal partitions. Simulation results on large systems with detailed generator and composite load models demonstrate that homogeneous scheduling complements Parareal, achieving nearly linear speedup while preserving MATE's expected performance. Consequently, the overall speedup of the hybrid solver approximates the product of the individual MATE and Parareal speedups. Heterogeneous scheduling offers performance benefits when uniform resource allocation is infeasible, allowing for flexible processor deployment across diverse computing environments.