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This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with exponential parametric distortion measurement errors. To estimate parameters in the distortion functions, we propose nonlinear least squares and weighted nonlinear least squares estimation methods under two identifiability conditions. After obtaining calibrated variables, the nonlinear least squares based estimators are proposed to estimate the parameters in the regression model. We studied the asymptotic results of estimators, especially we discuss the difference between the parametric calibrations and nonparametric calibrations. The latter is conducted as if the parametric structures in distortion functions are unknown. Simulation studies demonstrate the performance of the proposed estimators. [ABSTRACT FROM AUTHOR]