We consider finite element approximation of the parabolic obstacle problem. The analysis is based on a penalty formulation of the problem where the penalization parameter is allowed to vary in space and time. We estimate the penalization error in terms of the penalty parameter and the data of the equation. The penalized problem is discretized in space and time by means of a discontinuous Galerkin method. We prove an a posteriori error estimate in the space-time maximum norm involving a residual and the stability property of a linearized adjoint problem. The talk is based on joint work with D. A. French, R. H. Nochetto and M. Boman.