coupling the device to its input/output transmission line via a bandpass filter, leading to wider bandwidth at undiminished gain. Here we perform a non-perturbative circuit analysis in terms of dressed transmission-line modes for representative resonant coupling circuits, going beyond the weak-coupling treatment. The strong frequency dependence of the resulting coupling coefficients implies that the Markov approximation commonly employed in cQED analysis is inadequate. By retaining the full frequency dependence of the coupling, we arrive at a non-Markovian form of the quantum Langevin equation with the frequency-dependent complex-valued self-energy of the coupling in place of a single damping parameter. We also consistently generalize the input-output relations and unitarity conditions. Using the exact self-energies of elementary filter networks — a series- and parallel-LC circuit and a simple representative bandpass filter consisting of their combination — we calculate the generalized parametric gain factors. Compared with their Markovian counterpart, these gain profiles are strongly modified. We find bandwidth broadening not only in the established parameter regime, where the self-energy of the coupling is in resonance with the device and its real part has unity slope, but also within off-resonant parameter regimes where the real part of the self-energy is large. Our results offer insight for the bandwidth engineering of Josephson parametric devices using simple coupling networks.