Multimodal Continual Instruction Tuning (MCIT) aims to finetune Multimodal Large Language Models (MLLMs) to continually align with human intent across sequential tasks. Existing approaches often rely on the Mixture-of-Experts (MoE) LoRA framework to preserve previous instruction alignments. However, these methods are prone to Catastrophic Forgetting (CF), as they aggregate all LoRA blocks via simple summation, which compromises performance over time. In this paper, we identify a critical parameter inefficiency in the MoELoRA framework within the MCIT context. Based on this insight, we propose BranchLoRA, an asymmetric framework to enhance both efficiency and performance. To mitigate CF, we introduce a flexible tuning-freezing mechanism within BranchLoRA, enabling branches to specialize in intra-task knowledge while fostering inter-task collaboration. Moreover, we incrementally incorporate task-specific routers to ensure an optimal branch distribution over time, rather than favoring the most recent task. To streamline inference, we introduce a task selector that automatically routes test inputs to the appropriate router without requiring task identity. Extensive experiments on the latest MCIT benchmark demonstrate that BranchLoRA significantly outperforms MoELoRA and maintains its superiority across various MLLM sizes.