To study the cavity magnetic system and the coupling of magnons and phonons, especially the quantum sideband interference effect, we investigate the magnomechanically induced transparency and fast-slow light effects based on a hybrid cavity magnetic system with an Optical Parametric Amplifier (OPA) placed in it.We first give the composition of the hybrid cavity magnetic system. The Y3Fe5O12 (YIG) sphere is placed in the cavity mode, close to the maximum magnetic field of the cavity field, and in a uniform bias field. At the location of the YIG sphere, the magnetic field (along the x-axis), driving magnetic field (along the y-direction), and bias magnetic field (along the z-direction) of the cavity mode are perpendicular to each other. The system supports three different types of excitation, namely photons, magnetons, and phonons. A uniform bias field (z-direction) is applied to the YIG sphere to excite the interaction between the magnetic dipole and the magnon mode coupled by the cavity field. The magnetostrictive force causes YIG vibration, thus establishing a magnon-phonon interaction in the sphere. The microwave field is applied to enhance the coupling between magnons and phonons. An OPA is put in the cavity and driven by a driving laser. Next, according to the system model, we give the total Hamiltonian of the system, and the definitions or descriptions of each parameter are provided. After obtaining the Hamiltonian of the system in a frame rotating at the driving field frequency, the Heisenberg equations of motion for the operators are derived by using perturbation theory. We establish the relationship between input and output to acquire the amplitude and group delay expressions for the output field. Finally, we study magnomechanically induced transparency and fast-slow light effects using experimentally achievable parameters.Through numerical simulation, the results show that in the absence of OPA, there is a magnomechanically induced transparency window when only the cavity photon-magnon coupling is present. In the case where the magnon-phonon coupling strength K is not equal to zero, a double magnetically induced transparent window appears, and the dispersion curve becomes steeper at the position where the probe-driving detuning is equal to phonon frequency. As the coupling strength K increases, the depth of the transparent window deepens, the central peak value of the absorption spectrum curve widens, and the distance between the peaks on both sides also increases. Moreover, introducing OPA gain (G) leads to a more pronounced yellow area on the left side of the resonance frequency, indicating an asymmetry in the absorption spectrum. This is because the steady-state behavior of the cavity field is influenced by changes in OPA gain, which alters the phonon number in the system and leads to an asymmetrical absorption spectrum. The absorption peak value also gradually increases with the gain G. Maintaining G and K constant, as the cavity photon-magnon coupling increases, the absorption spectrum front at the resonance frequency becomes sharp, the peaks on both sides gradually widen, and the peak value does not change significantly. Further, by increasing the gain of OPA, it is found that the transmission rate at the position where the probe-driving detuning is equal to phonon frequency is greater than 1, thus realizing the regulation of the window transmission spectrum. The positive value of the curve on both sides of the system group delay increases, while the negative value decreases by enhancing G, indicating that choosing the appropriate OPA parameters can realize the switch between the low light and the fast light as well as enhance the slow light effect.In conclusion, we investigate the magnomechanically induced transparency and fast-slow light effects in a hybrid cavity magnetic system assisted by an OPA theoretically. The research results can provide a reference for the research of quantum optics manipulation and quantum information storage.