In automated design of turbomachinery, the method used to parametrize the blade shapes is key to ensure robustness and design flexibility of the entire optimization process. Typically, the two adopted methods are free-form deformation and computer-aided design based parametrization. The former allows for a large design flexibility, but it entails challenges to satisfy prescribed geometrical constraints. By contrast, the latter ensures a better control of the blade shape, but at the cost of narrowing down the design space. In this study, we aim at providing a unified methodology for a systematic assessment of the two methods by resorting to the adjoint method. A first comparison is conducted by carrying out shape optimization on a two-dimensional axial turbine cascade. The results indicate that, though the fluid-dynamic performance of the two optimal blade configurations are similar, the shapes are comparatively different, meaning that the optimization problem has multiple optima and that the type of parametrization is influencing the solution of the optimization problem.