Fourier decay in parabolic $C^{1+\alpha}$ systems with overlaps

We establish power Fourier decay for equilibrium states of parabolic $C^{1+\alpha}$ iterated function systems with overlaps satisfying a multiscale nonlinearity condition. This class includes the Lyons conductance measures $\nu_t$, $0<t<1$, associated to Galton-Watson trees with equal weights yielding advance towards a conjecture of Lyons on the absolute continuity of $\nu_t$ for small $t$. Further applications include Patterson-Sullivan measures for cusped hyperbolic surfaces, extending the work of Bourgain and Dyatlov to parabolic settings, conformal measures for Manneville-Pommeau and Lorenz-type maps, and the construction of the first genuinely $C^{1+\alpha}$ IFSs whose attractors have positive Fourier dimension but are not $C^1$-conjugate to linear IFSs. The proof combines the Bourgain-Dyatlov sum-product strategy with a multiscale induction approach that bypasses the use of spectral gaps for twisted transfer operators needed in several other works in the area.