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Asynchronous Federated Learning (AFL) has emerged as a significant research area in recent years. By not waiting for slower clients and executing the training process concurrently, it achieves faster training speed compared to traditional federated learning. However, due to the staleness introduced by the asynchronous process, its performance may degrade in some scenarios. Existing methods often use the round difference between the current model and the global model as the sole measure of staleness, which is coarse-grained and lacks observation of the model itself, thereby limiting the performance ceiling of asynchronous methods. In this paper, we propose FedPSA (Parameter Sensitivity-based Asynchronous Federated Learning), a more fine-grained AFL framework that leverages parameter sensitivity to measure model obsolescence and establishes a dynamic momentum queue to assess the current training phase in real time, thereby adjusting the tolerance for outdated information dynamically. Extensive experiments on multiple datasets and comparisons with various methods demonstrate the superior performance of FedPSA, achieving up to 6.37\% improvement over baseline methods and 1.93\% over the current state-of-the-art method.