One of the most computationally expensive problems in numerical linear algebra is the computation of the ε-pseudospectrum of matrices, that is, the locus of eigenvalues of all matrices of the form A + E, where ‖_E ‖ ≤ є. Several research efforts have been attempting to make the problem tractable by means of better algorithms and utilization of all possible computational resources. One common goal is to bring to users the power to extract pseudospectrum information from their applications, on the computational environments they generally use, at a cost that is sufficiently low to render these computations routine. To this end, we investigate a scheme based on i) iterative methods for computing pseudospectra via approximations of the resolvent norm, with ii) a computational platform based on a cluster of PCs and iii) a programming environment based on MATLAB enhanced with MPI functionality and show that it can achieve high performance for problems of significant size.