ABSTRACT Recent in situ and remote observations suggest that the transport regime associated with shock-accelerated particles may be anomalous i.e. the mean square displacement (MSD) of such particles scales non-linearly with time. We use self-consistent hybrid particle-in-cell plasma simulations to simulate a quasi-parallel shock with parameters compatible with heliospheric shocks, and gain insights about the particle transport in such a system. For suprathermal particles interacting with the shock we compute the MSD separately in the upstream and downstream regions. Tracking suprathermal particles for sufficiently long times up and/or downstream of the shock poses problems in particle plasma simulations, such as statistically poor particle ensembles and trajectory fragments of variable length in time. Therefore, we introduce the use of time-averaged mean square displacement (TAMSD), which is based on single-particle trajectories, as an additional technique to address the transport regime for the upstream and the downstream regions. MSD and TAMSD are in agreement for the upstream energetic particle population, and both give a strong indication of superdiffusive transport, consistent with interplanetary shock observations. MSD and TAMSD are also in reasonable agreement downstream, where indications of anomalous transport are also found. TAMSD shows evidence of heterogeneity in the diffusion properties of the downstream particle population, ranging from subdiffusive behaviour of particles trapped in the strong magnetic field fluctuations generated at the shock to superdiffusive behaviour of particles transmitted and moving away from the shock.