  Lightwave periodic driving of nearly dissipation-less currents has recently
emerged as a universal control concept for both superconducting (SC) and
topological electronics applications. While exciting progress has been made
towards THz-driven superconductivity, our understanding of the interactions
able to drive non-equilibrium pairing is still limited, partially due to the
lack of direct measurements of high-order correlation functions. Such
measurements would exceed conventional single-particle spectroscopies and
perturbative responses to fully characterize quantum states
far-from-equilibrium. Particularly, sensing of the exotic collective modes that
would uniquely characterize lightwave-driven SC coherence, in a way analogous
to the Meissner effect, is very challenging but much needed. Here we report the
discovery of lightwave-controlled superconductivity via parametric
time-periodic driving of the strongly-coupled bands in iron-based
superconductors by a unique phase-amplitude collective mode assisted by
broken-symmetry THz supercurrents. We are able to measure non-perturbative,
high-order correlations in this strongly-driven superconductivity by separating
the THz multi-dimensional coherent spectra into conventional pump-probe, Higgs
collective mode, and pronounced bi--Higgs frequency sideband peaks with highly
nonlinear field dependence. We attribute the drastic transition in the coherent
spectra to parametric excitation of time-dependent pseudo--spin canting states
modulated by a phase-amplitude collective mode that manifests as a strongly
nonlinear shift from $\omega_\mathrm{Higgs}$ to 2$\omega_\mathrm{Higgs}$.
Remarkably, the latter higher--order sidebands dominate over the lower-order
pump-probe and Higgs mode peaks above critical field, which indicates the
breakdown of the susceptibility perturbative expansion in the
parametrically-driven SC state.
