A new parallel numerical technique to estimate the effective elastic parameters of a rock core sample from the three-dimensional Computed Tomography images is presented. The method is based on the energy equivalence principle and a new approach to solving the 3D static elasticity problem by the iterative relaxation technique. The method in the three-dimensional case requires the obligatory parallel implementation. The most commonly used strategy of parallelization is MPI and OpenMP. The latest Fortran extension offers the new Coarray Fortran (CAF) features, which can potentially compete with the MPI due to its efficiency and simple implementation. We compare three parallel approaches based on the MPI, MPI+OpenMP and CAF to solve the problem. Comparison of these methods has shown that the CAF brings about a sufficiently compact parallel code with a simple syntax, thus making the parallelism easier to understand. The results presented demonstrate that the CAF implementation provides comparable performance to an equivalent MPI version.