Cross-market recommendation (CMR) faces severe challenges from distribution shifts between data-rich source markets and sparse target markets. Existing methods rely on a pre-training and fine-tuning paradigm for knowledge transfer, yet suffer from two key limitations: i) the objective gap between pre-training and full-parameter fine-tuning causes loss of generalized knowledge from source markets; ii) the high computational costs of extensive fine-tuning hinder scalability. To this end, we propose DCMPT, a novel Distilled Cross-Market Prompt-Tuning approach. DCMPT reframes the problem under a more efficient pre-training and prompt-tuning paradigm. Instead of full fine-tuning, we adapt a pre-trained universal backbone by freezing its weights and injecting a minimal set of learnable prompts to form a "student" model. To effectively optimize these prompts on sparse data, we introduce a novel teacher-student architecture: a specialized "teacher" model, trained exclusively on the target market, provides dense, market-specific supervision. This guidance is delivered via a dual distillation strategy designed to transfer global ranking patterns and adapt to local consumer tastes. Extensive experiments on real-world market datasets demonstrate that DCMPT significantly outperforms state-of-the-art methods, achieving superior target market performance with substantial parameter-efficiency. Code is provided in the supplementary material to ensure reproducibility.