This paper is concerned with the development of the method of multipliers combined with the conjugate gradient algorithm for solving the special linear systemwhere A is a real symmetric non negative definite (n×n) matrixBis a real (n×m) matrix with full column rank (m<n) and A and BThave no nontrivial null vector in common. We assume that A and BT are large and not extremely sparse. The proposed method is well suitable for parallel implementation on a multiprocessor system that can execute concurrently different tasks on a few vector processors with shared central memory, such as the CRAY Y-MP. The results of an extensive computer experimentation, which is aimed at evaluating the effectiveness of the method, are reported.