Prompt tuning offers a parameter-efficient way to adapt large pre-trained language models to new tasks, but most existing approaches are designed for single-task settings, failing to share knowledge across related tasks. We propose Cross-task Prompt Tuning (CrossPT), a modular framework for multi-task prompt tuning that enables controlled knowledge transfer while maintaining task-specific specialization. CrossPT decomposes each target prompt into shared, pre-trained source prompts and task-specific private prompts, combined via a learned attention mechanism. To support robust transfer, we systematically investigate key design factors including prompt initialization, balancing shared and private prompts, number of source prompts, learning rates, task prefixes, and label semantics. Empirical results on GLUE and related benchmarks show that CrossPT achieves higher accuracy and robustness compared to traditional prompt tuning and related methods, particularly in low-resource scenarios, while maintaining strong parameter efficiency.