A major achievement of the past decade has been the realization of macroscopic quantum systems by exploiting the interactions between optical cavities and mechanical resonators1-3. In these systems, phonons are coherently annihilated or created in exchange for photons. Similar phenomena have recently been observed through phonon-cavity coupling energy exchange between the modes of a single system mediated by intrinsic material nonlinearity4'5. This has so far been demonstrated primarily for bulk crystalline, high-quality-factor (Q > 105) mechanical systems operated at cryogenic temperatures. Here, we propose graphene as an ideal candidate for the study of such nonlinear mechanics. The large elastic modulus of this material and capability for spatial symmetry breaking via electrostatic forces is expected to generate a wealth of nonlinear phenomena6, including tunable intermodal coupling. We have fabricated circular graphene membranes and report strong phonon-cavity effects at room temperature, despite the modest Q factor (-100) of this system. We observe both amplification into parametric instability (mechanical lasing) and the cooling of Brownian motion in the fundamental mode through excitation of cavity sidebands. Furthermore, we characterize the quenching of these parametric effects at large vibrational amplitudes, offering a window on the all-mechanical analogue of cavity optomechanics, where the observation of such effects has proven elusive.