A parallel implementation of an unstructured finite volume method dealing with viscoelastic flows on a distributed computing environment through Parallel Virtual Machine (PVM) is reported in this paper. The numerical method is based on the SIMPLER algorithm, using a simplified PTT constitutive model and the Elastic Viscous Split Stress (EVSS) formulation. Unstructured triangular meshes with co-located arrangement are used, along with an equal-order interpolation method. This permits all the unknowns to be computed at the same grid points. The discretised algebraic equations are solved with an explicit Gauss-Seidel solver (point by point solver). The parallelisation of the program is implemented by a domain decomposition strategy using PVM platform. The program is tested on the viscoelastic flow past a cylinder between two parallel plates. Five sets of meshes are used and a good efficiency is achieved. It is found that the efficiency is strongly dependent on the grid size, the block number and the number of processors. The parallel efficiency increases with increasing problem size for a given block number. The domain decomposition efficiency decreases with increasing block number for a given problem size.