Compared to the large slope of the gain spectrum, in this paper, the characteristics of the slow and fast light on the band-edge of gain spectrum of the narrow-band fiber-optic parametric amplification (FOPA) are analyzed. Firstly, we derive the analytical solution of the signal delay under the condition of small-signal gain. Secondly, the characteristics of 10 Gb/s pseudo-random binary sequence pulse signal delay are investigated experimentally under the condition of low pump power (<1 W). Finally, the factors that influence the signal delay, such as pump power, dispersion, nonlinear coefficient, etc. are analyzed. The results obtained in this paper provide a detailed theoretical reference for designing slow and fast delay line in band-edge of gain spectrum of FOPA.