This paper addresses the problem of identifying and estimating the causal effect of a treatment in the presence of unmeasured confounding and various types of right-censoring. Examples of these censoring mechanisms are administrative censoring, competing risks and dependent censoring (e.g. loss to follow-up). Different parametric transformations are applied to each event time, resulting in a regression model with a more additive structure and error terms that are approximately normal and homoscedastic. The transformed event times are modeled using a joint regression framework, assuming multivariate Gaussian error terms with an unspecified covariance matrix. A control function approach is used to deal with unmeasured confounding. The model is shown to be identifiable and a two-step estimation procedure is proposed. This estimator is proven to yield consistent and asymptotically normal estimates. Furthermore, a goodness-of-fit test for the model's validity is developed. Simulations are conducted to examine the finite-sample performance of the proposed estimator under various scenarios. Finally, the methodology is applied to investigate the causal effect of job training programs on unemployment duration using data from the National Job Training Partnership Act (JTPA) study.