We investigate the strong approximation of stochastic parabolic partial differential equations with additive noise. We introduce postprocessing in the context of a standard Galerkin approximation, although other spatial discretizations are possible. In time, we follow [G. J. Lord and J. Rougemont, IMA J. Numer. Anal., 24 (2004), pp. 587–604] and use an exponential integrator. We prove strong error estimates and discuss the best number of postprocessing terms to take. Numerically, we evaluate the efficiency of the methods and observe rates of convergence. Some experiments with the implicit Euler–Maruyama method are described.