Parameter-efficient fine-tuning techniques like Low-Rank Adaptation (LoRA) have revolutionized the adaptation of large language models (LLMs) to diverse tasks. Recent efforts have explored mixtures of LoRA modules for multi-task settings. However, our analysis reveals redundancy in the down-projection matrices of these architectures. This observation motivates our proposed method, Mixture of Dyadic Experts (MoDE), which introduces a novel design for efficient multi-task adaptation. This is done by sharing the down-projection matrix across tasks and employing atomic rank-one adapters, coupled with routers that allow more sophisticated task-level specialization. Our design allows for more fine-grained mixing, thereby increasing the model’s ability to jointly handle multiple tasks. We evaluate MoDE on the Supernatural Instructions (SNI) benchmark consisting of a diverse set of 700+ tasks and demonstrate that it outperforms state-of-the-art multi-task parameter-efficient fine-tuning (PEFT) methods, without introducing additional parameters. Our findings contribute to a deeper understanding of parameter efficiency in multi-task LLM adaptation and provide a practical solution for deploying high-performing, lightweight models.