The study of survival data often requires taking proper care of the censoring mechanism that prohibits complete observation of the data. Under right censoring, only the first occurring event is observed: either the event of interest, or a competing event like withdrawal of a subject from the study. The corresponding identifiability difficulties led many authors to imposing (conditional) independence or a fully known dependence between survival and censoring times, both of which are not always realistic. However, recent results in survival literature showed that parametric copula models allow identification of all model parameters, including the association parameter, under appropriately chosen marginal distributions. The present paper is the first one to apply such models in a quantile regression context, hence benefiting from its well-known advantages in terms of e.g. robustness and richer inference results. The parametric copula is supplemented with a likewise parametric, yet flexible, enriched asymmetric Laplace distribution for the survival times conditional on the covariates. Its asymmetric Laplace basis provides its close connection to quantiles, while the extension with Laguerre orthogonal polynomials ensures sufficient flexibility for increasing polynomial degrees. The distributional flavour of the quantile regression presented, comes with advantages of both theoretical and computational nature. All model parameters are proven to be identifiable, consistent, and asymptotically normal. Finally, performance of the model and of the proposed estimation procedure is assessed through extensive simulation studies as well as an application on liver transplant data.