In this work we present address the combination of the Hierarchical Model (Hi-Mod) reduction approach with projection-based reduced order methods, exploiting either on Greedy Reduced Basis (RB) or Proper Orthogonal Decomposition (POD), in a parametrized setting. The Hi-Mod approach, introduced in, is suited to reduce problems in pipe-like domains featuring a dominant axial dynamics, such as those arising for instance in haemodynamics. The Hi-Mod approach aims at reducing the computational cost by properly combining a finite element discretization of the dominant dynamics with a modal expansion in the transverse direction. In a parametrized context, the Hi-Mod approach has been employed as the high-fidelity method during the offline stage of model order reduction techniques based on RB or POD. The resulting combined reduction methods, which have been named Hi-RB and Hi-POD, respectively, will be presented with applications in diffusion-advection problems, fluid dynamics and optimal control problems, focusing on the approximation stability of the proposed methods and their computational performance.