Cavity　optomechanical　systems　involving　an　optical　parametric　amplifier　(OPA)　can　exhibit　rich　classical　and　quantum　dynamical　behaviors.　By　simply　modulating　the　frequency　of　the　laser　pumping　the　OPA,　we　find　two　interesting　parameter　regimes,　with　one　of　them　enabling　us　to　study　quantum-classical　correspondence　in　system　dynamics,　while　there　exist　no　classical　counterparts　of　the　quantum　features　for　the　other.　For　the　detuning　of　the　laser　pumping　around　the　mechanical　frequency,　as　the　parametric　gain　of　OPA　increases　to　a　critical　value,　the　classical　dynamics　of　the　optical　or　mechanical　modes　can　experience　a　transition　from　the　regular　periodic　oscillation　to　a　period-doubling　motion,　in　which　case　the　light-mechanical　entanglement　can　be　well　studied　by　the　logarithmic　negativity　and　can　manifest　the　dynamical　transition　the　classical　nonlinear　dynamics.　In　addition,　the　optomechanical　entanglement　shows　a　second-order　transition　characteristic　at　the　critical　parametric　gain.　For　the　laser　detuning　being　about　twice　the　mechanical　frequency,　the　kind　of　normal　mode　splitting　comes　up　in　the　laser　detuning　dependence　of　optomechanical　entanglement,　which　is　induced　by　the　squeezing　of　the　optical　and　mechanical　hybrid　modes　and　finds　no　classical　correspondence.　The　OPA-assisted　optomechanical　systems　therefore　offer　a　simple　way　to　study　and　exploit　quantum　manifestations　of　classical　nonlinear　dynamics.