Fiber optic parametric and phase sensitive amplifiers (PSA) are interesting for modern day communication technologies due to their low noise and high gain amplification properties with a potential for all optical signal processing and wide band operation. PSAs are typically employed in either a single pump or dual pump configuration. In this article we explore the utilities of both configurations, however considering a fiber with a longitudinally varying dispersion profile. For the single pump case, PSA operation at large pump-signal detunings, that arise due to the longitudinal dispersion variation, were studied numerically, and recipes of using the system as a wide band wavelength selective filter were laid out. For the dual-pump case, emphasis was laid on achieving a larger signal gain, by reducing stimulated Brillouin scattering (SBS) that prevents large pump power transport through the nonlinear fiber. First, the effects of dispersion variation on the gain of a dual pump PSA were studied analytically and numerically in order to optimize the dispersion variation profile, neglecting SBS processes. Then we independently studied the SBS dynamics of the system numerically. A sinusoidally dispersion oscillating fiber (DOF) was found to be an optimal candidate with respect to its PSA and SBS performances. To establish this claim, we also experimentally compared the performance of an available DOF over a standard highly nonlinear fiber (HNLF) that has a constant dispersion profile and established its utility for designing a high gain PSA system, thanks to the SBS mitigation due to the longitudinal dispersion variation of the fiber.