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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 $ω_\mathrm{Higgs}$ to 2$ω_\mathrm{Higgs}$. 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.