Image copy-move forgery detection (CMFD) is crucial in image forensics. Copy-move forgeries have smaller feature differences than other forgeries, which makes them harder to detect. Nowadays, Fine-tuning pre-trained models to adapt CMFD is an effective transfer paradigm. It avoids the tedious process of designing specialized networks but consumes a lot of parameter resources. Meanwhile, the existing localization algorithms have low localization accuracy for multiple tampered areas. Therefore, we propose a new convolutional parameter adapter network (CPANet) to solve those problems. Firstly, a highly expressive convolutional adapter (C-adapter) is designed based on Prompt. It can learn the feature differences between the source domain and the target domain, and solve the problem of spatial heterogeneity in feature domains of different tasks. Secondly, Gradient-weighted Class Activation Mapping (Grad-CAM++) is introduced to realize the visual location of more than one suspicious target area using only gradient return without retraining the model. Finally, the C-adapter and Grad-CAM++ are integrated into the ConvNeXt to construct CPANet. Compared to full fine-tuning, CPANet reduces training parameters by more than 100 times. In the generalization performance experiment, the mean F1-score (F1) and the mean Area Under the ROC Curve (AUC) of CPANet are 80.09 and 0.8529, which are 7.28% and 5.63% higher than the current mainstream models. It is proved that the proposed CPANet has low consumption, high detection precision, and strong anti-attack characteristics.