Existing prompt-based fine-tuning methods typically learn task-specific prompts independently, imposing significant computing and storage overhead at scale when deploying multiple clinical natural language processing (NLP) systems. We present a multitask prompt distillation and decomposition framework that learns a single shared metaprompt from 21 diverse clinical source tasks and adapts it to unseen target tasks with fewer than 0.05% trainable parameters. Evaluated across five clinical NLP task types (named entity recognition, relation extraction, question answering, natural language inference, and summarization) on 10 held-out target datasets using three backbone models (LLaMA 3.1 8B, Meditron3 8B, gpt-oss 20B), our framework consistently outperforms LoRA by 1.5~1.7% despite using orders of magnitude fewer parameters, and exceeds single-task prompt tuning by 6.1~6.6%. The gpt-oss 20B model achieves the highest overall performance, particularly on clinical reasoning tasks. The strong zero- and few-shot performance demonstrates better transferability of the shared prompt representation.