In this talk a non-linearly coupled parabolic-elliptic problem is considered and we are interested in applying a trust-region reduced basis (RB) method for the parameter optimization. In this context, online-efficient projection-based error estimates are not available, and thus we develop hierarachical error estimators, which we are able to approximate by use of two different RB spaces. We show some results regarding efficiency and their use in a trust-region algorithm, where the two bases required to evaluate the etimator are built at the same time along the optimization path. Finally, we are going to see their use in an optimal input design problem, where the use of a trust-region optimization method is motivated by the necessity of executing the parameter optimization step multiple times.