We study the properties of optomechanically induced transparency and normal mode splitting in an optical parametric amplifier (OPA) cavity. We find that the value of phase of the field driving the OPA and the nonlinear gain of the OPA will evidently affect the width of the optomechanically induced transparency window and the property of the normal mode splitting. In particular, when the coupling field works at the mechanical red sideband, narrower transparency window than the case of empty cavity (no OPA) can be realized by adjusting the value of phase of the field driving the OPA and the nonlinear gain of the OPA, accompanied by the steep dispersion curve. These results may be applied to the realization of fast/slow light, and light storage in optomechanical system.