<p>We propose and theoretically investigate a scheme for the dynamic control of quantum entanglement in a hybrid system comprising an atom, an optical cavity, and a mechanical oscillator. By combining periodic optomechanical modulation with an ultrastrong atom-cavity interaction governed by the Rabi-Stark model, we derive an effective Hamiltonian and demonstrate programmable manipulation of both bipartite and tripartite entanglement. Our analysis identifies three key control parameters: the detuning, the atom-cavity nonlinear coupling, and the optomechanical coupling strength. We show that these parameters enable precise temporal shaping and pathway selection of quantum correlations. Specifically, we observe a time-divided, competitive distribution of entanglement between the atom-cavity and cavity-oscillator channels, where enhancing one often suppresses the other. The nonlinear coupling is found to extend entanglement death phases and trigger pulsed revivals, while stronger optomechanical coupling stabilizes the cavity-oscillator entanglement into a sustained plateau. These results establish a versatile framework for dynamically routing entanglement on demand, with direct applications for quantum memory, sensing beyond the standard quantum limit, and synchronization protocols in hybrid quantum networks.</p>