Acoustic Scene Classification (ASC) aims to identify the acoustic characteristics of different scenes. Acoustic data is collected by various devices in different cities, which leads to feature deviation of acoustic signals in the same scene and affects the classification performance of the model. Full fine-tuning model is a popular solution to adapt to ASC. It avoids the complexity of designing dedicated networks, while it consumes a lot of parameter resources. To solve this problem, we propose a multi-task efficient parameter tuning network (MT-EPTNet). First, we design a lightweight depthwise separable convolution adapter (Dsc-adapter) based on prompt. The Dsc-adapter is highly transferable, capable of learning feature differences between tasks and flexibly adapting to target tasks. Then, we integrate the Dsc-adapter into the speaker recognition model CAM++ to construct MT-EPTNet. By updating a small number of Dsc-adapter parameters, the frozen pre-trained model can efficiently perform on downstream audio tasks. The Dsc-adapter optimizes residual blocks and multi-granularity pooling blocks, enhances the invariance of MT-EPTNet to input feature shifts and more precisely captures local and global contextual features. Compared to full fine-tuning, MT-EPTNet reduces training parameters by 76.81%. It achieves an accuracy (Acc) of 83.6% on the benchmark ASC dataset, outperforming comparison models, and excels in other audio tasks. Experiments demonstrate that MT-EPTNet achieves high-accuracy ASC with low resource costs and exhibits excellent multi-task capabilities.