Current efforts toward a global energy transition are driving demand for new-generation, high-performance turbomachinery, often operating in challenging conditions or with unconventional fluids. In this context, new aerodynamic shape optimization (ASO) methods are becoming fundamental in the exploration of novel blade geometry designs. This research focuses on the optimization of parametric turbomachinery blade geometries through a data-driven ASO framework. While accurate, such approaches are often computationally prohibitive and, when focusing on industrial applicability, limit the admissible number of design parameters. This stems from the high number of high-fidelity Computational Fluid Dynamics (CFD) simulations needed during the exploration process. Surrogate models address these limitations by providing a low-fidelity, computationally inexpensive way to explore high-dimensional design spaces starting from a limited initial high-fidelity sampling. In the current work, Proper Orthogonal Decomposition (POD) surrogate models are integrated within an ASO workflow. POD is used to extract the principal flow features from a scarce number of initial design samples, then POD coefficients are interpolated to predict full flowfields in previously unseen design points, with a computational cost up to three orders of magnitude lower than a high-fidelity CFD simulation. To handle differing geometries, POD is coupled with a mesh deformation strategy based on Radial Basis Functions (RBFs), propagating the variations in blade geometry to surrounding mesh nodes continuously. In contrast to other methods, e.g. Free-Form Deformation, this strategy enables the control of mesh deformations by directly modifying the physically relevant design parameters. Finally, an optimization algorithm iterates over low-fidelity simulation data obtained by the POD to identify potential optimum sets of parameters, which are cross-validated with high-fidelity simulations. In summary, the proposed optimization framework reduces design costs and supports the development of advanced turbomachinery geometries, contributing to industrial and research efforts in decarbonization and energy efficiency.