Dopamine transporter (DAT) imaging is commonly used for monitoring Parkinson's disease (PD), where striatal DAT uptake amount is computed to assess PD severity. However, DAT imaging has a high cost and the risk of radiance exposure and is not available in general clinics. Recently, MRI patch of the nigral region has been proposed as a safer and easier alternative. This paper proposes a symmetric regressor for predicting the DAT uptake amount from the nigral MRI patch. Acknowledging the symmetry between the right and left nigrae, the proposed regressor incorporates a paired input-output model that simultaneously predicts the DAT uptake amounts for both the right and left striata. Moreover, it employs a symmetric loss that imposes a constraint on the difference between right-to-left predictions, resembling the high correlation in DAT uptake amounts in the two lateral sides. Additionally, we propose a symmetric Monte-Carlo (MC) dropout method for providing a fruitful uncertainty estimate of the DAT uptake prediction, which utilizes the above symmetry. We evaluated the proposed approach on 734 nigral patches, which demonstrated significantly improved performance of the symmetric regressor compared with the standard regressors while giving better explainability and feature representation. The symmetric MC dropout also gave precise uncertainty ranges with a high probability of including the true DAT uptake amounts within the range.