Post-translational modifications (PTMs) regulate protein function, stability, and interactions, playing essential roles in cellular signaling, localization, and disease mechanisms. Computational approaches enable scalable PTM site prediction; however, traditional models focus only on local sequence features from fragments around potential modification sites, limiting the scope of their predictions. Recently, pre-trained protein language models (PLMs) have improved PTM prediction by leveraging biological knowledge derived from extensive protein databases. However, most PLMs used for PTM site prediction are pre-trained solely on amino acid sequences, limiting their ability to capture the structural context necessary for accurate PTM site prediction. Moreover, these methods typically train separate single-task models for each PTM type, which hinders the sharing of common features and limits potential knowledge transfer across tasks. To overcome these limitations, we introduce MTPrompt-PTM, a multi-task PTM prediction framework developed by applying prompt tuning to a structure-aware protein language model (S-PLM). Instead of training several single-task models, MTPrompt-PTM trains one multi-task model to predict multiple types of PTM sites using shared feature extraction layers and task-specific classification heads. Additionally, we incorporate a knowledge distillation strategy to enhance the efficiency and generalizability of multi-task training. Experimental results demonstrate that MTPrompt-PTM outperforms state-of-the-art PTM prediction tools on 13 types of PTM sites, highlighting the advantages of multi-task learning and structural integration.