In this paper, we establish a model reduction technique for periodic discrete-time descriptor systems exploiting the generalized inverses of the periodic singular matrix pairs associated with the systems. We compute the generalized inverses of periodic singular matrix pairs to implement a structure preserving iterative method for the solution of the periodic projected Lyapunov equations that arise in analysis and modeling of periodic discrete-time descriptor systems. We extend the Smith method to solve the large scale projected periodic discrete-time algebraic Lyapunov equations in lifted form. A low-rank version of this method is also presented, which avoids the explicit lifted formulation and works directly with the period matrix coefficients. Moreover, we consider an application of the Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.