In this talk, we introduce a multilevel space-time multiplicative Schwarz method for solving parabolic equations in parallel. This approach treats the entire space-time system simultaneously, using the Schwarz method as a preconditioner for GMRES. We establish a convergence theory demonstrating that the method is optimal—its convergence rate is independent of key discretization and parallelization parameters. Large-scale parallel numerical results confirm this optimality and scalability, showing significant advantages over traditional time-stepping methods when thousands of processor cores are used.