A tumor growth model based on a parametric system of partial differential equations is considered. The system corresponds to a phenomenological description of a multi-species population evolution. A velocity field taking into account the volume increase due to cellular division is introduced and the mechanical closure is provided by a Darcy-type law. The complexity of the biological phenomenon is taken into account through a set of parameters included in the model that need to be calibrated. To this end, a system identification method based on a low-dimensional representation of the solution space is introduced. We solve several idealized identification cases corresponding to typical situations where the information is scarce in time and in terms of observable fields. Finally, applications to actual clinical data are presented.