This article is concerned with the discretization of parabolic optimization problems subject to pointwise in time constraints on mean values of the derivative of the state variable. Central components of the analysis are a priori error estimates for the dG(0)-cG(1) discretization of the parabolic partial differential equation (PDE) in the L-infinity(0,T;H-0(1)(Omega))-norm, together with corresponding estimates in L-1(0,T;H-0(1)(Omega)) for the adjoint PDE. These results are then utilized to show convergence orders for the discrete approximation toward the solution of the parabolic optimization problem.