We study the effect of nonparabolicity of the conduction band on the energy spectrum of the off-axis donors in InAs crater-like quantum rings in framework of the Kane Hamiltonian. By using ta simple numerical procedure we find the low-lying energies dependencies on the radius, height and width of the ring and on the distance of the donor location from the axis. The comparison of our results with those obtained in one-band effective mass approximation reveals an essential influence of the non-parabolicity of the conduction band on the off-axis energy spectra in InAs/GaAs quantum rings. In addition, we analyse the effect of the external magnetic field applied along the symmetry axis on the Aharonov-Bohm oscillations of lower energy levels.