Many previous studies have used the fitting of normal nonlinear regression models to analyze data with a symmetric distribution structure as nonlinear functions of unknown parameters. However, in practice, the distribution of residuals may be asymmetric and the choice of normal distribution is not appropriate. A family of statistical distributions that has recently received attention is the mixed-scaled normal distribution, which includes tailed and tail-weighted distributions such as tailed-t and tailed-slash as special cases. Since statistical inference of parameters by the marginal maximum likelihood method will lead to the solution of complex integrals with high dimensions, in this paper we use the Markov chain Monte Carlo simulation approach for Bayesian inference of model parameters. We also fit the nonlinear model with the proposed distributions on a set of real data to demonstrate the significance of the proposed model.