In structural mechanics, shape optimization is one way to improve mechanical properties of the structure. Typical objectives are to minimize displacements or to minimize stresses. In this context, it is desirable to conduct optimizations with parameterized, reduced order models. However, commercial finite element codes do not provide parameterized system matrices for a geometrical parameterization making an application of PMOR challenging. Therefore, a geometrically parameterized solid finite element is derived which can be formulated with respect to global design parameters. This leads to an affine representation of the system matrices. This allows an efficient application of interpolatory projection methods for parameterized systems. The approach is demonstrated with a numerical example where the proposed approach shows a significant speedup.