In this article, a parametric predictor is constructed to predict the future state of a discrete-time linear system with a known constant input delay and external disturbances by introducing a tuning parameter. Then, a predictor-based feedback controller is given to attenuate the unknown disturbance signal. The ultimate bound of the state of the resulted closed-loop system is obtained when the state feedback control law based on this novel predictor is utilized. Based on the expression of the ultimate bounds, the capacity to attenuate the unknown disturbance is analyzed for the parametric predictor within a certain range of parameters. An easily calculated interval is provided for the range of the tuning parameter such that the proposed predictor achieves a better disturbance attenuation capacity. Moreover, for the unknown constant disturbance, the estimation error between the proposed predictor with an arbitrary parameter and the future state can be eliminated. Finally, two numerical examples are utilized to illustrate the effectiveness of the proposed parametric predictor for bounded time-varying disturbances.