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Parkinson's disease (PD) is a common and complex neurodegenerative disorder with 5 stages in the Hoehn and Yahr scaling. Given the heterogeneity of PD, it is challenging to classify early stages 1 and 2 and detect brain function alterations. Functional magnetic resonance imaging (fMRI) is a promising tool in revealing functional connectivity (FC) differences and developing biomarkers in PD. Some machine learning approaches like support vector machine and logistic regression have been successfully applied in the early diagnosis of PD using fMRI data, which outperform classifiers based on manually selected morphological features. However, the early-stage characterization in FC changes has not been fully investigated. Given the complexity and non-linearity of fMRI data, we propose the use of a long short-term memory (LSTM) network to characterize the early stages of PD. The study included 84 subjects (56 in stage 2 and 28 in stage 1) from the Parkinson's Progression Markers Initiative (PPMI), the largest available public PD dataset. Under a repeated 10-fold stratified cross-validation, the LSTM model reached an accuracy of 71.63%, 13.52% higher than the best traditional machine learning method, indicating significantly better robustness and accuracy compared with other machine learning classifiers. We used the learned LSTM model weights to select the top brain regions that contributed to model prediction and performed FC analyses to characterize functional changes with disease stage and motor impairment to gain better insight into the brain mechanisms of PD.