A cavity optomechanical system containing an optical parametric amplifier is investigated under resolved sideband regime, which enhances the coupling between the movable mirror and the cavity field. The radiation-pressure inducing the movable mirror and the steady-state amplitude of the cavity field displaying an optical multistable behavior are studied. The modification of mechanical frequency and mechanical damping rate are analyzed induced by radiation pressure, which will change with the change of parametric gain, input laser power and parametric phase. In addition, the fluctuation spectrum of the movable mirror is also analyzed presenting the normal mode splitting with modulating the parametric phase of the driving field. Moreover, an accurate scheme is used to calculate the final effective mean phonon number that demonstrates the ground state cooling. The results show that the cooling of the mirror is dominated by the initial bath temperature, high mechanical quality factor and parametric phase. The parametric phase shows a new way to control the dynamics of nonlinear optomechanical cavity.