While data is distributed in multiple edge devices, Federated Learning (FL) is attracting more and more attention to collaboratively train a machine learning model without transferring raw data. FL generally exploits a parameter server and a large number of edge devices during the whole process of the model training, while several devices are selected in each round. However, straggler devices may slow down the training process or even make the system crash during training. Meanwhile, other idle edge devices remain unused. As the bandwidth between the devices and the server is relatively low, the communication of intermediate data becomes a bottleneck. In this paper, we propose Time-Efficient Asynchronous federated learning with Sparsification and Quantization, i.e., TEASQ-Fed. TEASQ-Fed can fully exploit edge devices to asynchronously participate in the training process by actively applying for tasks. We utilize control parameters to choose an appropriate number of parallel edge devices, which simultaneously execute the training tasks. In addition, we introduce a caching mechanism and weighted averaging with respect to model staleness to further improve the accuracy. Furthermore, we propose a sparsification and quantitation approach to compress the intermediate data to accelerate the training. The experimental results reveal that TEASQ-Fed improves the accuracy (up to 16.67% higher) while accelerating the convergence of model training (up to twice faster).