Various problems arising from physical processes are modelled using partial differential equations (PDEs). The Poisson's equation is the example of elliptic PDE, which we use for presenting the finite difference method for transforming the continuous problem to the discrete problem and obtaining sparse system of linear equations (SLEs). This paper discusses possibilities of parallel implementation of numerical methods for solving these special SLEs on clusters of workstations using message passing interface. We present theoretical background and evaluate results of our experiments done with OpenMPI library in parallel environment with which we have achieved reasonable high parallel efficiency for our problem. Superlinear speedup and constant efficiency for certain number of used processors have been achieved.