For a singularly perturbed parabolic equation on an interval, the first boundary value problem of reactiondiffusion type is studied. For the approximation of the boundary value problem we use previously developed finite difference schemes, of high $\varepsilon$-uniform order of accuracy in time, based on defect correction. The new approach developed in this paper is the introduction of a partitioning of the domain for these $\varepsilon$-uniform schemes. We determine conditions under which the difference schemes applied independently on the subdomains can accelerate ($\varepsilon$-uniformly) the solution of the boundary value problem without losing the accuracy of the original schemes. Hence, the simultaneous solution on the subdomains can in principle be used for parallelization of the computational method.