A fast computational technique is proposed to speed up the process of parametric macromodel extraction of Micro-Electro-Mechanical Systems (MEMS). An efficient technique of parametric macromodel extraction of MEMS is parametric model order reduction (PMOR). The key step and the main computational load of the popular parametric model order reduction methods is the computation of a projection matrix $V$ which requires computing moment matrices of the systems. For computing each moment matrix, the solution of a linear system with multiple right-hand sides is required. Usually, a considerable number of linear systems must be solved when the system includes more than two parameters. If the original system is of very large size, solving all the linear systems is the most computationally expensive step to obtain the reduced model. In this paper, a fast recycling algorithm GCRO-DR is applied to solve the whole sequence of linear systems. In addition, more efficient recycling algorithms G-DRvar1 and G-DRvar2 are proposed. Theoretical analysis and simulation results show that both the GCRO-DR and its variants G-DRvar1, G-DRvar2 are very efficient as compared with the standard solvers. Furthermore, the algorithms in this paper overcome the bottleneck of a recently proposed recycling method MKR-GMRES. By using the recycling algorithms, the PMOR process for extracting the macromodel can be significantly accelerated.