In fiber optical parametric amplifier, the nonlinear index induced by four-wave mixing effect varies with the frequency owing to the interaction between the strong pump wave and signal wave so that the group velocity of optical wave slows down or accelerates. Based on the analysis of parameters in the process of index change induced by the four-wave mixing in fiber, the influence of saturation effect of pump wave on the delay time of signal wave is investigated. By solving the nonlinear equation based on Runge-Kutta method, the results of analytical method and numerical method are compared when low power and high power signal waves are injected into the fiber, and the region with large difference is obtained. This region is located in the band edge of the spectrum of fiber optical parametric amplifier. These results are conductive to know the slow and fast light phenomenon in fiber optical parametric amplifier.