In this paper, we address the design of parameter-dependent mixed H2/Hinfin filters for output estimation of linear parameter-varying (LPV) plants that include a time-varying delay. We investigate the conditions to satisfy asymptotic stability, and H2 and Hinfin performance requirements in terms of linear matrix inequalities (LMIs). Our synthesis conditions, that provide mixed H2/Hinfin filters, are dependent linearly on the rate of change of the delay. We take the estimation error into account in the performance index and design the filters to minimize the output energy subject to a bound on the L2-gain from the noise to the estimation error. The designed filters are shown to have the capability of tracking the desired outputs of the plant in the presence of external disturbances and time varying delays. We examine two classes of filters: one that is rational and another one that has delay in the filter dynamics and, hence, it is nonrational. It is shown that the time-delayed (nonrational) filter results in reduced conservatism and improved performance. Illustrative examples are used to verify the design methodology and to demonstrate the superiority of using the proposed delayed filter configuration compared to the rational one.