Short-term metro inter-station flow forecasting plays a vital role in intelligent urban rail transit operation but remains challenging due to partial observability, data sparsity, and complex spatiotemporal dependencies. Multi-task learning (MTL) has emerged as a promising paradigm for jointly predicting multiple passenger flows. However, existing MTL-based approaches often suffer from limited parameter sharing across tasks and insufficient exploitation of common attributes, such as temporal patterns and spatial correlations between stations. To overcome these limitations, we propose MetroMMoE, a parameter-efficient short-term inter-station flow forecasting model based on the Multi-gate Mixture-of-Experts (MMoE) framework. MetroMMoE employs a customized MMoE layer to enable efficient mutual information sharing between inter-station origin-destination (OD) and destination-origin (DO) flow forecasting tasks, reducing computational and memory complexity of the task-interaction module from quadratic to linear compared to existing cross-attention mechanisms. Additionally, the model integrates learnable temporal embeddings and station positional embeddings to enhance feature discrimination and forecasting accuracy. Extensive experiments on two real-world metro datasets from Shanghai and Hangzhou demonstrate that MetroMMoE consistently outperforms state-of-the-art baselines in both OD and DO flow forecasting, achieving superior accuracy with nearly 50% fewer parameters than the leading HIAM.