Domain wall resonance spectra in the weakly nonlinear regime for garnet films with a perpendicular anisotropy supporting parallel stripe domains have been investigated using micromagnetic simulations and zero-field broadband ferromagnetic resonance experiments. The main characteristics of the 2D numerical micromagnetic approach we developed is to solve the Landau-Lifshitz equation by an iterative method in the frequency domain and to incorporate a nonlinear phenomenological damping term. It is shown that the nonlinear damping affects simultaneously the driving field dependencies of the resonance frequency and the resonance linewidth for the fundamental domain wall resonance of parallel stripe domains, and the critical field for the domain wall resonance foldover. The micromagnetic simulations allow us to reproduce quantitatively both the nonlinear redshift of the domain wall resonance frequency and the nonlinear line broadening experimentally observed for increasing values of the input microwave power.