Space-time FEM, stage-parallel implicit Runge Kutta-methods, and, in general, parallel-in-time methods offer promising solutions to the limited scalability of algorithms on distributed systems when parallelism is only exploited in the spatial domain. These algorithms cannot run faster than a certain threshold even if more hardware resources are added. Identifying additional parallelism might be a solution. [1] presented the implementation of a stage-parallel multi-level preconditioner for implicit Runge-Kutta methods, using multigrid, and demonstrated that the scaling limit can indeed be shifted. This talk expands on the concept of time parallelism by adopting the idea of (stage-parallel) multilevel preconditioners to tensor-product space-time finite-element discretizations. The method is facilitated by the multigrid infrastructure and matrix-free capabilities of the deal.II library. It addresses both high-order continuous and discontinuous variational time discretizations. We demonstrate its effectiveness for the heat and Stokes equations by comparing it to a space-time multigrid method [2, 3]. [1] P. Munch, I. Dravins, M. Kronbichler, M. Neytcheva, Stage-parallel fully implicit Runge-Kutta implementations with optimal multilevel preconditioners at the scaling limit, SIAM Journal on Scientific Computing (SISC), vol. 46(2), pp. S71–S96, 2024. [2] N. Margenberg, P. Munch, A space-time multigrid method for space-time finite element discretizations of parabolic and hyperbolic PDEs, arXiv:2408.04372, 2024. [3] N. Margenberg, M. Bause, P. Munch, An hp multigrid approach for tensor-product space-time finite element discretizations of the Stokes equations, SIAM Journal on Scientific Computing, vol. 47(6), pp. B1503-1529, 2025.