In this paper we consider the implementation of particle tracking Monte Carlo on two different types of parallel computer architectures, namely, (i) the AMT DAP-610 array of processors which is a SIMD machine and (ii) the transputer based Meiko Computing Surface which is a MIMD machine. An analogue, research-level, fixed source, particle transport Monte Carlo code for studying the attenuation and leakage of gamma ray photons in simple, multilayer, shielding configurations, originally written for serial computers, is modified and re-written to run efficiently on these two different types of parallel machine. The philosophy adopted and algorithms developed in transferring the code to the parallel machines are described. Two illustrative problems are solved using realistic cross-section data, involving a 9 MeV source of gamma photons in a lead-void-water sphere and slab. Integral quantities, such as fraction of particles absorbed and escaped, and differential quantities, such as flux distributions and leakage spectra, computed by the parallel codes, are presented in tables and graphs. For equivalent calculations, the CPU times on the DAP and a number of serial computers (e.g. ICL-3900, CRAY XMP/28 in scalar mode, SUN 4, and VAX 11/750) are compared, and the resulting speedup factors quoted. For the Meiko Computing Surface, the performance obtained from running the code when the number of transputers used is varied, is tabulated. Finally, the performance of the two parallel machines is compared.