Identification of elastic and plastic properties of materials from indentation tests received considerable attention in the open literature. However, unambiguous and automatic determination of parameters in the case of the crystal plasticity (CP) model is still an unsolved problem. In this paper, we investigate the possibility to unambiguously identify the CP parameters from spherical indentation tests using finite element method simulations combined with evolutionary algorithm (EA). To this aim, we check the efficiency and accuracy of EA while fitting either load–penetration curves, surface topographies, or both at the same time. By fitting the results against simulation data with known parameters, we can verify the accuracy of each parameter independently. We conclude that the best option is to fit both load–penetration curve and surface topography at the same time. To understand why a given fitting scheme leads to correct values for some parameters and incorrect values for others, a sensitivity analysis was performed.