Many model order reduction methods for parameterized systems need to construct a projection matrix V which requires computing several moment matrices of the parameterized systems. For computing each moment matrix, the solution of a linear system with multiple right-hand sides is required. Furthermore, the number of linear systems increases with both the number of moment matrices used and the number of parameters in the system. Usually, a considerable number of linear systems has to be solved when the system includes more than two parameters. The standard way of solving these linear systems in case sparse direct solvers are not feasible is to use conventional iterative methods such as GMRES or CG. In this paper, a fast recycling algorithm is applied to solve the whole sequence of linear systems and is shown to be much more efficient than the standard iterative solver GMRES as well as the newly proposed recycling method MKR-GMRES from. As a result, the computation of the reduced-order model can be significantly accelerated.