The main parallelisation related features of a computational tool based on the finite element method (FEM) for the numerical modelling of mineral-slurry like flows are described in this work. In particular, both the domain decomposition method (DDM) and the processes communication strategy employed are discussed in detail. The DD algorithm is based on the iterative update of the boundary conditions imposed on the interfaces between subdomains, the so-called transmission conditions. Due to its versatility in several parallel architectures, the message-passing standard used here is the message passing interface (MPI) one. Since mineral-slurries rheology may change according to the prevailing local flow conditions, Newtonian and non-Newtonian viscous fluids are considered in this work. Indeed, both Newtonian and non-Newtonian laminar flows are numerically studied in two well-known canonical configurations usually found in mineral-slurry transport. The main results show that the parallel FEM based tool is capable of carrying out high-fidelity numerical simulations of mineral-slurry like flows. Finally, in all numerical simulations performed, relatively good speedups were obtained.