Complex real-life structures can exhibit multiple types of nonlinearities including geometric, material and contact nonlinearities. Their behavior is modelled by using nonlinear high fidelity Finite Element (FE) models, generally consisting of a high number of degrees of freedom due to complex geometric features. This leads to high computational loads that can be reduced by the application of model order reduction (MOR) methods. A large set of projection based MOR methods of nonlinear structural systems are described in literature. These methods are generally based on static/dynamic snapshots and eigenmodes that are condensed into a reduced basis (e.g. MEM method [1]). Further speedup is achieved through hyper reduction [1]. In addition, variable model parameters can be accounted for in tailored parametric MOR (pMOR) methods [2]. The use of one global basis, switching between local bases or the interpolation of parametric bases among others are options to create a parameter dependent basis. Geometrically and material nonlinear structural systems in rolling contact pose challenges in the development of pMOR methods due to their combination of distributed nonlinearity and local contact nonlinearity with a high number of potential contact locations around the circumference. The resulting high number of snapshots needed to account for this makes a global basis approach unfeasible. Instead, a local basis that exploits the rotational symmetry of the structure is employed to efficiently adjust the basis according to the rotational angle. The parametrization and inclusion of rigid body rotations in the basis are also required. A total lagrangian nonlinear FE tire model as an example of such a rolling contact nonlinear system is presented. It is implemented in an in-house Matlab nonlinear FE toolbox. The pMOR methods of basis switching and basis interpolation are analyzed with respect to the displacement and the contact solution in comparison to the full order model solution.