  Non-linear interactions between collective modes play a definitive role in
far out of equilibrium dynamics of strongly correlated electron systems.
Understanding and utilizing these interactions is crucial to photo-control of
quantum many-body states. One of the most surprising examples of strong mode
coupling is the interaction between apical oxygen phonons and Josephson
plasmons in bilayer YBa$_2$Cu$_3$O$_{6+x}$ superconductors. Experiments by Hu
et al (2014). and Kaiser et al. (2014) showed that below Tc, photo-excitation
of phonons leads to enhancement and frequency shifts of Josephson plasmon
edges, while aboveTc, photo-excited phonons induce plasmon edges even when
there are no discernible features in the equilibrium reflectivity spectrum.
Recent experiments by Van Hoegen et al. (2019) also observed parametric
generation of Josephson plasmons from photo-excited phonons both below Tc and
in the pseudogap phase. In this paper we present a theoretical model of
phonon-plasmon three wave interaction arising from coupling between the oxygen
motion and the in-plane superfluid stiffness. Analysis of the parametric
instability of plasmons based on this model gives frequencies of the most
unstable plasmons that are in agreement with experimental observations. We also
discuss how strong parametric excitation of Josephson plasmons can explain pump
induced changes in the TeraHertz reflectivity of YBa$_2$Cu$_3$O$_{6+x}$ in the
superconducting state, including frequency shifts and sharpening of Josephson
plasmon edges, as well as appearance of a new peak around 2THz. An interesting
feature of this model is that overdamped Josephson plasmons do not give any
discernible features in reflectivity in equilibrium, but can develop plasmon
edges when parametrically excited. We suggest that this mechanism explains
photo-induced plasmon edges in the pseudogap phase of YBa$_2$Cu$_3$O$_{6+x}$.
