In this paper, we propose a new parallel bidirectional algorithm, based on Cholesky factorization, for the solution of sparse symmetric system of linear equations. Unlike the existing algorithms, the numerical factorization phase of our algorithm is carried out in such a manner that the entire back substitution component of the substitution phase is replaced by a single step division. Since there is a substantial reduction in the time taken by the repeated execution of the substitution phase, our algorithm is particularly suited for the solution of systems with multiple b- vectors. The effectiveness of our algorithm is demonstrated by comparing it with the existing parallel algorithm, based on Cholesky factorization, using extensive simulation studies on two-dimensional problems discretized by FEM.