Assuming the availability of a hardware system that permits ideal N-fold concurrent processing, a new parallel algorithm for the numerical integration of ordinary differential equations is proposed. The algorithm is based on the predictor and corrector formulas and their iterative use. The reliability and convergence properties of the solutions are studied by performing the simulations on a conventional computer system for some specific examples. It is found that the proposed algorithm is indeed a practical one and offers much greater computing speed than the existing algorithms.