In this work we present both industrial and biomedical applications, focusing on shape parametrization and parameter space reduction by means of active subspaces. In particular we introduce a combined parameter and model reduction methodology using a POD-Galerkin approach, and its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids . The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid artery. A parametric description of the admissible deformations, based on radial basis functions interpolation technique implemented in the PyGeM python package, is introduced. The use of the reduced order model acting on the reduced parameter space allows significant computational savings and better performances. Moreover we present the reduction of heterogeneous parameter space in a naval engineering problem, that is the hydrodynamic flow past the hull of a ship advancing in calm water , considering structural and shape parameters. The geometrical parametrization is done via free form deformation. Some perspectives on a complete shape optimization pipeline by means of Dynamic Mode Decomposition (DMD) and POD with interpolation (PODI) are presented , together with the integration of the python packages PyDMD and EZyRB respectively.