In this paper a modified parallel Jacobi-conditioned conjugate gradient (CG) method is proposed for solving linear elastic finite element system of equations. The conventional element-by-element and diagonally conditioned approaches are discussed with respect to parallel implementation on distributed memory MIMD architectures. The effects of communication overheads on the efficiency of the parallel CG solver are considered and it is shown that for the efficient performance of a parallel CG solver, the interprocessor communication has to be carried out concurrently. A concurrent communication scheme is proposed by relating the semi-bandwidth of the stiffness matrix with the number of independent degrees of freedom and the number of processors and inducing directionalization of communication within the processor pipeline. With the aid of two examples the effectiveness of the proposed method is demonstrated showing that the cost of communication remains low and relatively insensitive to the increase in the number of processors.