An equally spaced series of very large and nearly constant-voltage self-induced singularities has been observed in the dc I-V characteristics of a small Josephson tunnel junction strongly coupled to a resonant section of a superconducting transmission line. The system allows extremely high values of the coupling parameter. The current steps are due to subharmonic parametric excitation of the fundamental mode of the resonator loaded by the junction admittance. Using an applied magnetic field to vary the coupling parameter, we traced out half-integer steps as well as the mode steps known from more weakly coupled systems. Based on a single-resonator model, we explain the exceptional size of the steps by the large content of higher harmonics of the Josephson oscillation sustained by the multiple modes of the resonator. The parameters of the junction and the loaded resonator—measured in situ by using the junction as detector—form a consistent set. The dominant system loss originates in the junction but cannot be ascribed to a relevant conductance.