Multimodal large language models (MLLMs) enable powerful cross-modal reasoning capabilities but impose substantial computational and latency burdens, posing critical challenges for deployment on resource-constrained edge devices. In this paper, we propose MSAO, an adaptive modality sparsity-aware offloading framework with edge-cloud collaboration for efficient MLLM Inference. First, a lightweight heterogeneous modality-aware via fine-grained sparsity module performs spatial-temporal-modal joint analysis to compute the Modality Activation Sparsity (MAS) metric, which quantifies the necessity of each modality with minimal computational overhead. Second, an adaptive speculative edge-cloud collaborative offloading mechanism dynamically schedules workloads between edge and cloud based on the derived MAS scores and real-time system states, leveraging confidence-guided speculative execution to hide communication latency. Extensive experiments on VQAv2 and MMBench benchmarks demonstrate that MSAO achieves a 30% reduction in end-to-end latency and 30%-65% decrease in resource overhead, while delivering a throughput improvement of 1.5x to 2.3x compared to traditional approaches, all without compromising competitive accuracy.

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