We deal with the fundamental problem of path following applied to control mobile manipulators. A parametric-based path following algorithm is proposed. Such an approach results in a cascaded structure of the control system, so that the control algorithm is designed using the backstepping integrator method. The proposed solution is robust due to the following features. Firstly, it is based on the Bishop parametrization, which is well-defined at every point of the curve. Moreover, we present a novel approach to the orthogonal projection method onto the path so that the motion precisely along the path is possible. Finally, the robustness to structural and parametric uncertainties of the dynamics model is guaranteed thanks to the sliding mode controller applied at the dynamic level of the control cascade. The problem is analyzed theoretically. The achieved results are verified with an exemplary simulation study. The proposed algorithm assures asymptotic convergence of errors to zero for less strict requirements imposed on the desired path and in the case of partial knowledge of the dynamics model.