This note is concerned with the problem of parity space-based fault estimation for linear discrete time-varying systems. The design of parity space-based fault estimator is formulated as to find a minimum for a matrix quadratic form. It is shown that the minimum provides a unified solution to the fault estimation problem with different design criteria, such as the performance indices of maximum singular value and Frobenius-norm. A necessary and sufficient condition for the minimum is derived and an analytic solution to the parity space-based fault estimator is obtained. Numerical examples are given to show the effectiveness of the proposed method.