Parametric amplification enables sensitive microwave detection and signal processing by amplifying weak signals with minimal added noise. Here, we demonstrate kinetic inductance mediated parametric amplification in a hybrid magnonic system comprising a nonlinear NbN nanowire resonator strongly coupled to a yttrium iron garnet sphere. The resonator exhibits four wave mixing and nonlinear frequency pulling, enabling parametric amplification both away from and within the magnon-photon strong coupling regime. In the magnon detuned regime, we observe 7 dB gain, while coupling to the YIG magnon shifts the parametric amplification response by ~20 MHz and yields 3.4 dB gain in the hybrid mode. Thus, magnon photon coupling provides a route to magnetically tune the spectral position of kinetic inductance parametric amplification. At higher pump powers, the system transitions to a self oscillatory regime featuring comb-like sidebands and probe activated nonlinear mode mixing. These results establish hybrid magnonics as a platform for tunable nonlinear microwave amplification and dynamics.