The conjugate gradient method is applied to a large, sparse, highly structured linear system of equations obtained from a finite difference discretization of the Poisson equation. The matrix-free implementation of the matrix-vector product is shown to be optimal with respect to both memory usage and performance. The parallel implementation of the method can give excellent performance on a cluster of workstations, with the optimal number of processors depending on the quality of the interconnect hardware. This justifies the use of the method as computational kernel for the time-stepping in a system of reaction-diffusion equations.