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Narrowband fiber parametric amplification is known to provide slow and fast light capabilities. Here, an analytical expression of the maximum slow light time delay achievable in ideal, isotropic fibers is derived. Then, the effects of random birefringence on the slow and fast light ability are numerically investigated by integrating the governing equations over a large number of statistical realizations of the fiber. The different polarization rotation along the fiber for the signal, the idler, and the pump reduces the amplifier mean gain and the mean time delay. For small random birefringence, the decrease of the mean delay can be directly imputed to the reduction of the mean gain. For large random birefringence, severe pulse distortion occurs, the mean delay is further reduced and the delay uncertainty highly enhanced. The influence of the stimulated Raman scattering is finally addressed.