Parallel computing is a powerful way to leverage modern computer architecture to reduce the simulation time of differential equation systems. Dividing the spatial domain enables space-parallelism, but scaling this approach will reach a saturation limit. Thereby, the overhead in communication between the subdomains will prohibit or even reverse further speed-up. Additionally, time-parallelism can be introduced to scale beyond this saturation limit and efficiently utilize the hardware of current high-performance compute architectures. In our work, we coupled the two open-source frameworks: LibPFASST, which handles parallel-in-time integration, and AMReX, which serves as a space-parallel solver [1, 2]. The combination of space and time parallelization, as well as the separate handling by different frameworks, can be natively explained via spectral deferred corrections - the backbone of the PFASST algorithm [3, 4]. PFASST divides the full time-interval into smaller ones, and introduces parallelism between those. Additionally, we extend LibPFASST with ”parallel across the method”-ideas, enabling parallel time-integration in each subinterval [5, 6]. The developed framework is used to solve the advection-diffusion problem, a two-compartment flow model, and the inviscid Burgers equations in varying spatial dimensions. Based on these problems, the benefits of parallel-in-time integration as an extension of space-parallelism are highlighted, and additionally, pure space-, space-time-, and space-time2 -parallelization are also benchmarked against each other.