  The possibility of enhancing desirable functional properties of complex
materials by optical driving is motivating a series of studies of their
nonlinear terahertz response. In high-Tc cuprates, large amplitude excitation
of certain infrared-active lattice vibrations has been shown to induce
transient features in the reflectivity suggestive of non-equilibrium
superconductivity. Yet, a microscopic mechanism for these observations is still
lacking. Here, we report measurements of time- and scattering-angle-dependent
second-harmonic generation in YBa$_2$Cu$_3$O$_{6+x}$, taken under the same
excitation conditions that result in superconductor-like terahertz
reflectivity. We discover a three-order-of-magnitude amplification of a
2.5-terahertz electronic mode, which is unique because of its symmetry,
momentum, and temperature dependence. A theory for parametric three-wave
amplification of Josephson plasmons, which are assumed to be well-formed below
T$_c$ but overdamped throughout the pseudogap phase, explains all these
observations and provides a mechanism for non-equilibrium superconductivity.
More broadly, our work underscores the role of parametric mode mixing to
stabilize fluctuating orders in quantum materials.
