We generalize asymptotic properties obtained in the observation-driven times series models considered by [R. Douc, P. Doukhan, and E. Moulines, Ergodicity of observation-driven time series models and consistency of the maximum likelihood estimator, Stochastic Processes Appl., 123(7):2620–2647, 2013] in the sense that the conditional law of each observation is also permitted to depend on the parameter. The existence of ergodic solutions and the consistency of the maximum likelihood estimator (MLE) are derived under easy-to-check conditions. The obtained conditions appear to apply for a wide class of models. We illustrate our results with specific observation-driven times series, including the recently introduced NBIN-GARCH and NM-GARCH models, demonstrating the consistency of the MLE for these two models.