In this technical note we provide a smoothed perturbation analysis (SPA) estimator of the sensitivity of a discrete time Parisian option with respect to the barrier level. The analysis put forward is of interest in a broader context than that of exotic options as we provide an SPA analysis for a problem where the critical event for the SPA estimator is based on an entire sample path, which is a novelty in the literature. Numerical examples illustrate the performance of the estimator.