Bounded response variables are common in many applications where the responses are percentages, proportions, or rates. New regression models have been proposed recently to model the relationship among one or more covariates and the conditional mean of a response variable based on the beta distribution or a mixture of beta distributions. However, when we are interested in knowing how covariates impact different levels of the response variable, quantile regression models play an important role. A new quantile parametric mixed regression model for bounded response variables is presented by considering the distribution introduced by [27]. A Bayesian approach is adopted for inference using Markov Chain Monte Carlo (MCMC) methods. Model comparison criteria are also discussed. The inferential methods can be easily programmed and then easily used for data modeling. Results from a simulation study are reported showing the good performance of the proposed inferential methods. Furthermore, results from data analyses using regression models with fixed and mixed effects are given. Specifically, we show that the quantile parametric model proposed here is an alternative and complementary modeling tool for bounded response variables such as the poverty index in Brazilian municipalities, which is linked to the Gini coefficient and the human development index.