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Hybrid spin-mechanical setups offer a versatile platform for quantum science and technology, but improving the spin-phonon as well as the spin-spin couplings of such systems remains a crucial challenge. Here, we propose and analyze an experimentally feasible and simple method for exponentially enhancing the spin-phonon, and the phonon-mediated spin-spin interactions in a hybrid spin-mechanical setup, using only \emph{ linear resources}. Through modulating the spring constant of the mechanical cantilever with a time-dependent pump, we can acquire a tunable and nonlinear (two-phonon) drive to the mechanical mode, thus amplifying the mechanical zero-point fluctuations and directly enhancing the spin-phonon coupling. Our method allows the spin-mechanical system to be driven from the weak-coupling regime to the strong-coupling regime, and even the ultrastrong coupling regime. In the dispersive regime, this method gives rise to a large enhancement of the phonon-mediated spin-spin interaction between distant solid-state spins, typically\emph{two orders of magnitude larger} than that without modulation. As an example, we show that the proposed scheme can apply to generating entangled states of multiple spins with high fidelities even in the presence of large dissipations.