Numerical solution of 3D linear elasticity equations is considered. Problem is described by a coupled system of second order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements. Preconditioned Conjugate Gradient iterative method is used for solving large-scale linear algebraic systems arising after the Finite Element Method (FEM) discretization of the problem. The displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block- diagonal part of the original matrix. Then circulant block- factorization is used to precondition thus obtained block- diagonal matrix. Since both preconditioning techniques, displacement decomposition and circulant block- factorization, are highly parallelizable, a portable parallel FEM code based on MPI is developed. Results of numerical tests performed on a number of modern parallel computers using real-life engineering problems from the geomechanics in geosciences are reported and discussed.