The impact of random dispersion fluctuations on the gain and saturation behavior of a one-pump fiber optical parametric amplifier (1-P FOPA) in the presence of fourth-order dispersion coefficient (β4) is investigated. Three coupled amplitude equations with fiber losses are solved numerically for the calculation of the pump, signal, and idler. The performances of 1-P FOPA are also analyzed with variation of dispersion fluctuation amplitude (σ) and correlation length (Lc). Based on the numerical results, it is found that the gain spectra and saturation curves exhibit some differences when β4 is considered. In comparison with the case where β4 is ignored, the peak gain remains the same, but the 3 dB bandwidth increases when β4 exists. Another notable difference is that the saturation power is shifted to lower or higher values, depending on the σ and Lc parameters. In general, the peak gain reduces as σ increases, and the peak gain reduction is greater for the case of shorter Lc. The numerical analysis is probably useful, especially for the case where the signal wavelength is detuned far from the pump wavelength.