Quantile regression allows us to estimate the relationship between covariates and any quantile of the response variable rather than the mean. Recently, several statistical distributions have been considered for quantile modeling. The objective of this study is to provide a new computational package, two biomedical applications, one of them with COVID-19 data, and an up-to-date overview of parametric quantile regression. A fully parametric quantile regression is formulated by first parameterizing the baseline distribution in terms of a quantile. Then, we introduce a regression-based functional form through a link function. The density, distribution, and quantile functions, as well as the main properties of each distribution, are presented. We consider 18 distributions related to normal and non-normal settings for quantile modeling of continuous responses on the unit interval, four distributions for continuous response, and one distribution for discrete response. We implement an R package that includes estimation and model checking, density, distribution, and quantile functions, as well as random number generators, for distributions using quantile regression in both location and shape parameters. In summary, a number of studies have recently appeared applying parametric quantile regression as an alternative to the distribution-free quantile regression proposed in the literature. We have reviewed a wide body of parametric quantile regression models, developed an R package which allows us, in a simple way, to fit a variety of distributions, and applied these models to two examples with biomedical real-world data from Brazil and COVID-19 data from US for illustrative purposes. Parametric and non-parametric quantile regressions are compared with these two data sets.