In this work, a new penalty formulation is proposed for the analysis of Mindlin-Reissner plates by using the element-free Galerkin method. A penalized weak form for the Mindlin-Reissner Plates is constructed through the exterior penalty method to enforce the essential boundary conditions of rotations as well as transverse displacements. In the numerical examples, some typical problems of Mindlin-Reissner plates are analyzed, and parametric studies on the order of integration and the size of influence domain are also carried out. The effect of the types of background cells on the accuracy of numerical solutions is observed and a proper type of background cell for obtaining optimal accuracy is suggested. Further, optimal order of integration and basis order of Moving Least Squares approximation are suggested to efficiently handle the irregularly distributed nodes through the triangular type of background cells. From the numerical tests, it is identified that unlike the finite element method, the proposed element-free Galerkin method with penalty technique gives highly accurate solution without shear locking in dealing with Mindlin-Reissner plates.