A domain decomposition procedure for parabolic equations is described. In this procedure, the computational domain is divided into nonoverlapping subdomains. The equation is discretized by finite differences in time, and in space, a Galerkin finite element method is used on each subdomain. Subdomain solutions are related by an explicit flux calculation on the interfaces between subdomains. The interface fluxes are calculated in a stable and accurate manner, thus no iterations between the interface and subdomains are required. The method has been implemented on an Intel iPSC/860 Hypercube, and comparisons between domain decomposition solutions and a fully implicit Galerkin solution are presented for a set of test problems.