The literature of predicting the outstanding liability for insurance companies has undergone rapid and profound changes in the past three decades, most recently focusing on Bayesian stochastic modeling and multivariate insurance loss payments. In this article, we introduce a novel Bayesian multivariate model based on the use of parametric copula to account for dependencies between various lines of insurance claims. We derive a full Bayesian stochastic simulation algorithm that can estimate parameters in this class of models. We provide an extensive discussion of this modeling framework and give examples that deal with a wide range of topics encountered in the multivariate loss prediction settings.