Many applications in photonics require all-optical manipulation of plasma waves, 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 weakly damped plasma waves, involving oscillatory tunneling of the superfluid between capacitively coupled planes. Such Josephson plasma waves (JPWs) are also highly nonlinear, and exhibit striking phenomena like cooperative emission of coherent terahertz radiation, superconductor-metal oscillations and soliton formation. We show here that terahertz JPWs in cuprate superconductors can be parametrically amplified through the cubic tunneling nonlinearity. 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 fluctuations or single terahertz-photon devices.