This paper is the second part of our work on a priori error analysis for finite element discretizations of parabolic optimal control problems. In the first part [SIAM J. Control Optim., 47 (2008), pp. 1150-1177] problems without control constraints were considered. In this paper we derive a priori error estimates for space-time finite element discretizations of parabolic optimal control problems with pointwise inequality constraints on the control variable. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. For the treatment of the control discretization we discuss different approaches, extending techniques known from the elliptic case.