  We present a novel approach to the parallelization of the parabolic fast
multipole method for a space-time boundary element method for the heat
equation. We exploit the special temporal structure of the involved operators
to provide an efficient distributed parallelization with respect to time and
with a one-directional communication pattern. On top, we apply a task-based
shared memory parallelization and SIMD vectorization. In the numerical tests we
observe high efficiencies of our parallelization approach.
