This paper is concerned with an approximation scheme for rate-independent systems governed by a non-smooth dissipation and a possibly non-convex energy functional. The scheme is based on the local minimization scheme introduced in [EM06], but relies on local stationarity of the underlying minimization problem. Under the assumption of Mosco-convergence for the dissipation functional, we show that accumulation points exist and are so-called parametrized solutions of the rate-independent system. In particular, this guarantees the existence of parametrized solutions for a rather general setting. Afterwards, we apply the scheme to a model for the evolution of damage.