Adapting Large Language Models (LLMs) to a continuous stream of tasks is a critical yet challenging endeavor. While Parameter-Efficient Fine-Tuning (PEFT) methods have become a standard for this, they face a fundamental dilemma in continual learning. Reusing a single set of PEFT parameters for new tasks often leads to catastrophic forgetting of prior knowledge. Conversely, allocating distinct parameters for each task prevents forgetting but results in a linear growth of the model's size and fails to facilitate knowledge transfer between related tasks. To overcome these limitations, we propose a novel adaptive PEFT method referred to as \textit{Mixtures of SubExperts (MoSEs)}, a novel continual learning framework designed for minimal forgetting and efficient scalability. MoSEs integrate a sparse Mixture of SubExperts into the transformer layers, governed by a task-specific routing mechanism. This architecture allows the model to isolate and protect knowledge within dedicated SubExperts, thereby minimizing parameter interference and catastrophic forgetting. Crucially, the router can adaptively select and combine previously learned sparse parameters for new tasks, enabling effective knowledge transfer while ensuring that the model's capacity grows sublinearly. We evaluate MoSEs on the comprehensive TRACE benchmark datasets. Our experiments demonstrate that MoSEs significantly outperform conventional continual learning approaches in both knowledge retention and scalability to new tasks, achieving state-of-the-art performance with substantial memory and computational savings.