While powerful in image-conditioned generation, multimodal large language models (MLLMs) can display uneven performance across demographic groups, highlighting fairness risks. In safety-critical clinical settings, such disparities risk producing unequal diagnostic narratives and eroding trust in AI-assisted decision-making. While fairness has been studied extensively in vision-only and language-only models, its impact on MLLMs remains largely underexplored. To address these biases, we introduce FairLLaVA, a parameter-efficient fine-tuning method that mitigates group disparities in visual instruction tuning without compromising overall performance. By minimizing the mutual information between target attributes, FairLLaVA regularizes the model's representations to be demographic-invariant. The method can be incorporated as a lightweight plug-in, maintaining efficiency with low-rank adapter fine-tuning, and provides an architecture-agnostic approach to fair visual instruction following. Extensive experiments on large-scale chest radiology report generation and dermoscopy visual question answering benchmarks show that FairLLaVA consistently reduces inter-group disparities while improving both equity-scaled clinical performance and natural language generation quality across diverse medical imaging modalities. Code can be accessed at https://github.com/bhosalems/FairLLaVA.