While many ideas and proofs of concept for parallel-in-time integration methods exists, the number of large-scale, accessible time-parallel codes is rather small. This is often due to the apparent or subtle complexity of the algorithms and the many pitfalls awaiting developers of parallel numerical software. One example of such a time-parallel code is pySDC, which implements, among others, the parallel full approximation scheme in space and time (PFASST). Inspired by nonlinear multigrid ideas, PFASST allows to integrate multiple time steps simultaneously using a space-time hierarchy of spectral deferred corrections. In this paper, we demonstrate the application of performance analysis tools to the PFASST implementation pySDC. We trace the path we took for this work, show examples of how the tools can be applied, and explain the sometimes surprising findings we encountered. Although focusing only on a single implementation of a particular parallel-in-time integrator, we hope that our results and in particular the way we obtained them are a blueprint for other time-parallel codes.