Large Vision-Language Models (LVLMs) have demonstrated impressive performance on multimodal tasks through scaled architectures and extensive training. Recent studies introduce Mixture of Experts (MoE) into LVLMs for improved computational efficiency. However, existing MoE approaches treat visual and linguistic modalities with symmetric architectures, overlooking the inherent asymmetry in how these two modalities are processed. This asymmetry causes two critical issues. First, text and vision form hierarchical rather than parallel relationships, as text queries typically describe partial aspects of complete visual scenes. Euclidean expert space struggles to encode such containment structures. Second, language experts in deeper layers progressively shift from evidence-based processing to parametric memory dependence, losing grounding in the provided visual and linguistic information. To address these issues, we propose AsyMoE, a novel architecture that explicitly models this asymmetry through three specialized expert groups. Intra-modality experts handle modality-specific processing. Hyperbolic inter-modality experts capture hierarchical cross-modal relationships through negative curvature geometry. Evidence-priority language experts suppress parametric memory activation and maintain contextual grounding throughout network depth. Extensive experiments demonstrate that AsyMoE achieves consistent improvements over baseline methods, with average gains of 1.5\% over MoE variants and up to 3.8\% on hallucination-sensitive tasks. AsyMoE activates 25.45\% fewer parameters compared to dense models.