Automatic taxonomy completion aims to attach the emerging concept to an appropriate pair of hypernym and hyponym in the existing taxonomy. Existing methods suffer from the overfitting to leaf-only problem caused by imbalanced leaf and non-leaf samples when training the newly initialized classification head. Besides, they only leverage subtasks, namely attaching the concept to its hypernym or hyponym, as auxiliary supervision for representation learning yet neglect the effects of subtask results on the final prediction. To address the aforementioned limitations, we propose TacoPrompt, a Collaborative Multi-Task Prompt Learning Method for Self-Supervised Taxonomy Completion. First, we perform triplet semantic matching using the prompt learning paradigm to effectively learn non-leaf attachment ability from imbalanced training samples. Second, we design the result context to relate the final prediction to the subtask results by a contextual approach, enhancing prompt-based multi-task learning. Third, we leverage a two-stage retrieval and re-ranking approach to improve the inference efficiency. Experimental results on three datasets show that TacoPrompt achieves state-of-the-art taxonomy completion performance. Codes are available at https://github.com/cyclexu/TacoPrompt.