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Cavity-mediated magnon-magnon coupling can lead to a transfer of spin-wave excitations between two spatially separated magnetic samples. In this paper, we present a scheme to substantially amplify this transfer efficiency through the application of a two-photon parametric drive on the cavity. Further, we enunciate the physical origin of the enhancement. The recurrent multiphoton absorption by the cavity opens up an infinite ladder of accessible energy levels, which can induce higher-order transitions within the magnon Fock space. This is reflected in a heightened spin-current response from one of the magnetic samples when the neighboring sample is coherently pumped. The enhancement induced by the parametric drive can be considerably high within the stable dynamical region. Specifically, near the periphery of the stability boundary, the spin current is amplified by several orders of magnitude. Such striking …