Abstract Flexible distributional regression models where regression predictors are related to not only the mean but also other distributional parameters are quite common nowadays. However, such model specifications are usually restricted to univariate response variables although the joint analyses of multiple responses is often of considerable interest and allows one to provide additional insights, especially related to the dependence between conditionally interrelated responses. The main reason for the notable absence of such model specifications is, arguably, the difficulty in setting up flexible yet interpretable multivariate distributions for the response vector. This is even more the case when considering response vectors that comprise discrete as well as continuous response components. Motivated by a case study on child health in developing countries, we propose copula-based bivariate mixed binary-continuous regression models for the simultaneous analysis of a continuous indicator of acute undernutrition and a binary indicator for the presence/absence of fever for children in India. Utilising the latent continuous representation of binary regression models, we develop fully Bayesian inference for the resulting class of copula regression models where the latent continuous responses are imputed as an additional step in the Markov chain Monte Carlo simulations. We study the performance of the resulting model in our application and show that the analysis benefits both from the flexible specification of regression effects including nonlinear effects of continuous covariates and spatial effects and from the simultaneous analysis of two response variables.