A new parallel algorithm for the numerical integration of ordinary differential equations is proposed. The algorithm is based on the predictor and corrector formulae and their iterative use. An equation which describes the error propagation of the proposed algorithm is derived and discussed. The reliability and convergence properties of the solutions are studied by performing the numerical experiments on a conventional computer system for some specific examples. It is found that the proposed algorithm is indeed a practical one for a concurrentt system such as a multiprocessor and a vector processor, and offers much greater computing speed than the existing algorithms.