Many phenomena in science and engineering are governed by evolutionary partial differential equations. In some cases, such as three-dimensional models or long-time simulations, the computational cost becomes a bottleneck for obtaining fast and accurate approximations of the solution. To exploit the capabilities of modern parallel computing architectures, significant research is being conducted in the field of space-time parallel methods. In this talk, we introduce a new class of space-time parallel solvers that combines the iterative multigrid-reduction-in-time (MGRIT) strategy with suitable time-splitting integrators, enabling non-iterative spatial parallelization. We present the main theoretical results and demonstrate the scalability and performance of the proposed approach through a series of numerical experiments.