The use of 3D Computed Tomography (CT) throws light on the interior structure of the core samples. That is why the 3D CT has become increasingly widespread in digital rock physics. However, the CT does not provide information about the elastic properties of the rocks. So far, the most common approach to recover these properties is static laboratory measurements, when a sample is subject to a variety of static loads, followed by the measurement of displacements. This way is rather expensive, time-consuming and sometimes result in the destruction of a sample. Instead, we propose to implement numerical experiments by solving a 3D elastic static problem to perform upscaling of the CT data and to compute effective elastic parameters. In fact, to find these parameters we need to resolve a huge system of linear algebraic equations, which is a troublesome task. We have decided not to use a direct solver, but to apply a new iterative relaxation technique by considering a dynamic elastic problem with a special choice of relaxation parameters. The approach proposed needs parallelization. We have come to the conclusion to use the new feature of the latest Fortran extension known as CoArray Fortran (CAF) and to compare the three ways of parallel implementation as applied to this problem: MPI, MPI+OpenMP and CAF. Our experiments have proved that CAF and MPI approximately demonstrate the same performance, but the CAF possesses a clearer structure and is easy for coding.