In this paper we show how to construct parallel explicit multistep algorithms for an accurate and efficient numerical integration of the radial Schrödinger equation. The proposed methods are adapted to Bessel functions, that is to say, they integrate exactly any linear combination of Bessel and Newman functions and ordinary polynomials. They are the first of the like methods that can achieve any order. The coefficients of the method are computed in each step.We show how the parallel implementation of the method is the key of an efficient computation.