High-overtone bulk acoustic wave resonators (HBARs) promise advanced phononics, yet achieving nonlinearity remains challenging. We demonstrate a radiation-pressure-type parametric interaction between GHz HBARs and low-frequency flexural modes in a suspended silicon nitride membrane, where mechanical displacement modulates the external dissipation rate to enable dissipative acousto-mechanical coupling. Benefiting from the high quality factor, the system enters the resolved-sideband regime at room temperature, yielding acousto-mechanically induced transparency. We observe tunable Kerr nonlinearity and generate coherent HBAR frequency combs via two-tone driving. Notably, our dissipative coupling strength is 20 times larger than the dispersive coupling, the highest ratio among reported hybrid dissipative-dispersive coupling systems, resulting in the experimental observation of amplification in the reflection spectra under red-sideband driving. The ability to interface dense HBAR modes with a common mechanical resonator provides a scalable on-chip platform for multimode phononic information processing, with quantum phononics potentially achievable at sub-Kelvin temperatures.