Dissipation is generally regarded as detrimental to nonclassical-state generation, yet whether it can determine which quantum resource is emitted from an open hybrid quantum system remains largely unexplored. Here, we show that the symmetry of bosonic dissipation channels can select and switch quantum-emission pathways in a tripartite Mollow system, where a strongly driven two-level emitter simultaneously couples to cavity photons and motional phonons. In the Mollow regime, higher-order tripartite interactions generate resonant photon-phonon pair excitations, whose emitted quantum states are subsequently selected by dissipation. Under symmetric dissipation, the system produces strongly correlated photon-phonon pairs, enabling both single-pair emission and correlated multipair bundles with pronounced nonclassical correlations. Breaking the dissipation symmetry progressively suppresses the hybrid correlations and dynamically decouples the photonic and phononic degrees of freedom. In the strongly asymmetric regime, long-lived phonons accumulate in highly excited Fock states, while rapidly leaking cavity photons are converted into high-purity single-photon and two-photon bundles. The crossover between these distinct emission regimes is characterized through time-resolved correlation functions, Fock-state distributions, and Wigner quasiprobability distributions. Our results establish dissipation symmetry as a resource for steering quantum emission between hybrid and purely photonic nonclassical states, providing a route toward programmable hybrid quantum emitters and dissipative quantum-state engineering.