An efficient L 0-stable parallel algorithm is developed for the two-dimensional diffusion equation with non-local time-dependent boundary conditions. The algorithm is based on subdiagonal Pade approximation to the matrix exponentials arising from the use of the method of lines and may be implemented on a parallel architecture using two processors running concurrently with each processor employing the use of tridiagonal solvers at every time-step. The algorithm is tested on two model problems from the literature for which discontinuities between initial and boundary conditions exist. The CPU times together with the associated error estimates are compared.