To extend prevailing scaling limits when solving time-dependent partial differential equations, the parallel full approximation scheme in space and time (PFASST) has been shown to be a promising parallel-in-time integrator. Similar to a space-time multigrid, PFASST is able to compute multiple time-steps simultaneously and is therefore in particular suitable for large-scale applications on high performance computing systems. In this work we couple PFASST with a parallel spectral deferred correction (SDC) method, forming an unprecedented doubly time-parallel integrator. While PFASST provides global, large-scale "parallelization across the step", the inner parallel SDC method allows to integrate each individual time-step "parallel across the method" using a diagonalized local Quasi-Newton solver. This new method, which we call "PFASST with Enhanced concuRrency" (PFASST-ER), therefore exposes even more temporal parallelism. For two challenging nonlinear reaction-diffusion problems, we show that PFASST-ER works more efficiently than the classical variants of PFASST and can be used to run parallel-in-time beyond the number of time-steps.