A generalized strongly regular graph of grade , as a new generalization of strongly regular graphs, is a regular graph such that the number of common neighbours of both any two adjacent vertices and any two non-adjacent vertices takes on distinct values. For any vertex of a generalized strongly regular graph of grade 2 with parameters , if the number of the vertices that are adjacent to and share common neighbours with , or are non-adjacent to and share common neighbours with is independent of the choice of the vertex , then the generalized strongly regular graph of grade 2 is free. In this paper, we investigate the generalized strongly regular graph of grade 2 with parameters and provide the sufficient and necessary conditions for the existence of a family of free generalized strongly regular graphs of grade 2.