A 360-area surface-based cortical parcellation is extended to study mild cognitive impairment (MCI) and Alzheimer's disease (AD) from healthy control (HC) using the joint human connectome project multi-modal parcellation (JHCPMMP) proposed by us. We propose a novel classification method named as JMMP-LRR to accurately identify different stages toward AD by integrating the JHCPMMP with the logistic regression-recursive feature elimination (LR-RFE). In three-group classification, the average accuracy is 89.0% for HC, MCI, and AD compared to previous studies using other cortical separation with the best classification accuracy of 81.5%. By counting the number of brain regions whose feature is in the feature subset selected with JMMP-LRR, we find that five brain areas often appear in the selected features. The five core brain areas are Fusiform Face Complex (L-FFC), Area 10d (L-10d), Orbital Frontal Complex (R-OFC), Perirhinal Ectorhinal (L-PeEc) and Area TG dorsal (L-TGd, R-TGd). The features corresponding to the five core brain areas are used to form a new feature subset for three classifications with the average accuracy of 80.0%. Results demonstrate the importance of the five core brain regions in identifying different stages toward AD. Experiment results show that the proposed method has better accuracy for the classification of HC, MCI, AD, and it also proves that the division of brain regions using JHCPMMP is more scientific and effective than other methods.