Throughout this work, we aim to explore the linear optical properties of a semiconductor multi-shell spherical quantum dot with and without a hydrogenic donor impurity. The core and well layers are defined by the parabolic electronic potentials in the radial direction. The energy levels and corresponding wavefunctions of the structure are calculated by using the shooting technique in the framework of the effective-mass approximation. We investigate the intersublevel absorption coefficients of a single electron and the hydrogenic donor impurity comparatively as a function of the photon energy. In addition, we carry out the effect of a donor impurity and the layer thickness on the oscillator strengths and magnitude and position of absorption coefficient peaks. We illustrate the electron probability distribution and variation of the energy levels in cases with and without the impurity for different thicknesses of layers. This kind of structure gives an opportunity to tune and control the absorption coefficient of the system by changing three different thickness parameters. Also it provides a possibility to separate 0s and 1p electrons in different regions of the quantum dot.