This talk focuses on the theoretical foundations of a new parallel-in-space-and-time solver for parabolic problems. I will begin by introducing our two-level Schwarz preconditioner, built for GMRES and BDF2 discretization. The core of my presentation will be the rigorous proof of its optimal convergence. Specifically, I will demonstrate that the convergence rate is independent of key parameters like the mesh size and the number of subdomains. I will then present supporting numerical evidence that verifies both the robustness and accuracy of our approach.