In Part I of this paper, we proposed a new parallel bidirectional algorithm, based on Cholesky factorization, for the solution of sparse symmetric system of linear equations. In this paper, we propose a new parallel bidirectional algorithm, based on LU factorization, for the solution of general sparse system of linear equations having non-symmetric coefficient matrix. As with the sparse symmetric systems, 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. However, due to absence of symmetry, important differences arise in the ordering technique, the symbolic factorization phase, and message passing during numerical factorization phase. The bidirectional substitution phase for solving general sparse systems is the same as that for sparse symmetric systems. The effectiveness of our algorithm is demonstrated by comparing it with the existing parallel algorithm, based on LU factorization, using extensive simulation studies.