Many applications in photonics require all-optical manipulation of plasma waves1, which can concentrate electromagnetic energy on sub-wavelength length scales. This is difficult in metallic plasmas because of their small optical nonlinearities. Some layered superconductors support Josephson plasma waves (JPWs)2,3, involving oscillatory tunneling of the superfluid between capacitively coupled planes. Josephson plasma waves are also highly nonlinear4, and exhibit striking phenomena like cooperative emission of coherent terahertz radiation5,6, superconductor-metal oscillations7 and soliton formation8. We show here that terahertz JPWs can be parametrically amplified through the cubic tunneling nonlinearity in a cuprate superconductor. Parametric amplification is sensitive to the relative phase between pump and seed waves and may be optimized to achieve squeezing of the order parameter phase fluctuations9 or single terahertz-photon devices.