In my talk, I will present results on implementations of fully stage-parallel solvers for Radau-IIA implicit Runge-Kutta (IRK) time integrators. The application context is partial differential equations discretized by higher-order finite element methods, run on large-scale parallel computers where partitioning only the spatial domain might not offer enough parallelism to saturate the available computing resources. My talk will present key algorithms for the parallel execution of IRK methods, including the use of tensor product formulations to expose parallel operations. Two variants will be considered, one approximating the inverse of the Runge-Kutta matrix by a lower-triangular matrix as well as a direct diagonalization of the Runge-Kutta matrix. While the latter involve fewer solver iterations, the use of complex arithmetic reduces the available parallelism. The opportunities and limitations of these methods using a set of different implementation options against traditional time integrators will be assessed for the model problem of the heat equation. My work relies on recent contributions to the widely used deal.II finite element library for cutting-edge ingredients for the spatial part, including multigrid preconditioners and matrix-free operator evaluation for optimal node-level performance.