Particle-In-Fourier (PIF) schemes are attractive for long-time integration of kinetic plasma simulations as they conserve charge, momentum and energy (up to time discretization error), exhibit a variational structure, do not have aliasing and have excellent stability properties. However, they are typically more expensive than the commonly used Particle-In-Cell (PIC) schemes due to the requirement of non-uniform discrete Fourier transforms (DFT) or fast Fourier transforms (FFT). In this talk, we propose a Parareal-based parallel-in-time integration method for PIF schemes by employing a PIF scheme of coarser tolerance for nonuniform FFTs or the standard PIC scheme as coarse propagators towards the goal of performing long-time integration simulations. We perform an error analysis of the algorithm and verify the results numerically with Landau damping, two-stream instability, and Penning trap test cases in 3D-3V. We also implement the space-time parallelization of the PIF schemes in the open-source, performance-portable library ``Independent Parallel Particle Layer" (IPPL) and conduct massively parallel scaling studies on JUWELS and JEDI supercomputers with A100 and GH200 GPUs. The space-time parallelization provides up to $4-6$ times speedup compared to spatial parallelization alone and achieves a push rate of more than 1 billion particles per second for the benchmark plasma mini-apps considered.