Mixture cure rate models have been developed to analyze failure time data where a proportion never fails. For such data, standard survival models are usually not appropriate because they do not account for the possibility of non-failure. In this context, mixture cure rate models assume that the studied population is a mixture of susceptible subjects who may experience the event of interest and non-susceptible subjects that will never experience it. More specifically, mixture cure rate models are a class of survival time models in which the probability of an eventual failure is less than one and both the probability of eventual failure and the timing of failure depend (separately) on certain individual characteristics. In this paper, we propose a Bayesian approach to estimate parametric mixture cure rate models with covariates. The probability of eventual failure is estimated using a binary regression model, and the timing of failure is determined using a Weibull distribution. Inference for these models is attained using Markov Chain Monte Carlo methods under the proposed Bayesian framework. Finally, we illustrate the method using data on the return-to-prison time for a sample of prison releases of men convicted of sexual crimes against women in England and Wales and we use mixture cure rate models to investigate the risk factors for long-term and short-term survival of recidivism.