Within mathematical research, Geometric Topology deals with the study of piecewise-linear n-manifolds, i.e. triangulable spaces which appear locally as the $n$-dimensional Euclidean space. This paper reports on the computational aspects of an algorithm for generating triangulations of PL 3- and 4-manifolds represented by edge-coloured graphs. As the number of graph vertices is increased the algorithm becomes computationally expensive very quickly, making it a natural candidate for the usage of HPC resources. We present an optimized, parallel version of the algorithm that is suitable for deployment of multi-core systems. Scalability results are discussed on two different platforms, namely an IBM iDataPlex Linux cluster and the IBM supercomputer BlueGene/Q.