The synthesis of ultra-long lived acoustic phonons in a variety of materials and device geometries could enable a range of new coherent information processing and sensing technologies; many forms of phonon dissipation pose a barrier to this goal. We explore linear and nonlinear contributions to phonon dissipation in silica at cryogenic temperatures using fiber-optic structures that tightly confine both photons and phonons to the fiber-optic core. When immersed in helium, this fiber system supports nearly perfect guidance of 9 GHz acoustic phonons; strong electrostrictively mediated photon-phonon coupling (or guided-wave stimulated Brillouin scattering) permits a flexible form of laser-based phonon spectroscopy. Through linear and nonlinear phonon spectroscopy, we isolate the effects of disorder-induced two-level tunneling states as a source of phononic dissipation in this system. We show that an ensemble of such two-level tunneling states can be driven into transparency-virtually eliminating this source of phonon dissipation over a broad range of frequencies. Experimental studies of phononic self-frequency saturation show excellent agreement with a theoretical model accounting for the phonon coupling to an ensemble of two-level tunneling states. Extending these results, we demonstrate a general approach to suppress dissipation produced by two-level tunneling states via cross-saturation, where the lifetime of a phonons at one frequency can be extended by the presence of a high intensity acoustic beam at another frequency. Our modeling and measurements suggest that Rayleigh scattering dominates phonon losses for the longest lifetimes achieved in our system. Although these studies were carried out in silica, our findings are quite general, and can be applied to a range of materials systems and device geometries.