In signal restoration problems, we expect to improve the restoration performance with a priori information about unknown target signals. In this paper, the parametric Wiener filter with linear constraints for unknown target signals is discussed. Since the parametric Wiener filter is usually defined as the minimizer of the criterion not for the unknown target signal but for the filter, it is difficult to impose constraints for the unknown target signal in the criterion. To overcome this difficulty, we introduce a criterion for the parametric Wiener filter defined for the unknown target signal whose minimizer is equivalent to the solution obtained by the original formulation. On the basis of the newly obtained criterion, we derive a closed-form solution for the parametric Wiener filter with linear constraints.