In this paper we provide an a priori error analysis for parabolic optimal control problems with a pointwise (Dirac-type) control in space on three-dimensional domains. The two-dimensional case was treated in [D. Leykekhman and B. Vexler, SIAM J. Numer. Anal., 51 (2013), pp. 2797-2821]; however, the three-dimensional case is technically much more involved. To approximate the problem we use standard continuous piecewise linear elements in space and the piecewise constant discontinuous Galerkin method in time. Despite low regularity of the state equation, we establish O(root k + h) order of convergence rate for the control in the L-2 norm. This result improves almost twice the previously known estimate in [W. Gong, M. Hinze, and Z. Zhou, SIAM J. Control Optim., 52 (2014), pp. 97-119] and in addition does not require any relationship between the time step k and the mesh size h. The main technical tools are discrete maximal parabolic regularity results and the best approximation-type estimate for the finite element error in the L-infinity(Omega; L-2(I)) norm.