We study a quantum many-body variant of the parametric oscillator by investigating the driven sine-Gordon model with a modulated tunnel coupling via a semiclassical truncated Wigner approximation (TWA). We first analyze the parametric resonant regime for driving protocols that retain our model gapped, and compare the TWA to a time-dependent Gaussian variational ansatz (TGVA). We then turn to a drive which closes the gap, resulting in an enhanced energy absorption. While the TGVA approach breaks down in this regime, we can apply TWA to explore the dynamics of the mode-resolved energy density and the higher-order correlations between modes in the prethermal heating regime. For weak driving amplitude, we find an exponentially fast energy absorption in the main resonant mode, while the heating of all remaining modes is almost perfectly suppressed on short timescales. At later times, the highly excited main resonance provides effective resonant driving terms for its higher harmonics through the nonlinearities in the Hamiltonian, and gives rise to an exponentially fast heating in these particular modes. We capture the strong correlations induced by these resonant processes by evaluating higher-order connected correlation functions. Our results can be experimentally probed in ultracold-atomic settings, with parallel one-dimensional quasicondensates in the presence of a modulated tunnel coupling.