This contribution presents reduced order modelling techniques for the efficient solution of parametric problems on spline discretizations. The problems of interest are formulated on parameterized geometries, where the domain and solution discretization depend on geometrical parameters. The goal is to construct efficient reduced order models for fast parametric simulations within a real-time and many-query context, which suits perfectly the demands of typical optimization workflows. In particular, tailored localization strategies enable efficient and accurate reduction for problems with moving discontinuities that stem from the underlying discretization or physical problem at hand. These strategies are based on clustering of solution snapshots and allow an efficient offline/online decomposition, while the online cost involved in solving local reduced problems is low. Numerical examples on spline discretizations demonstrate the accuracy and efficiency of the reduced order modelling techniques for linear elastic model problems.