Accurate medical image segmentation is essential for effective diagnosis and treatment. Previously we proposed PraNet-V1 as a means to enhance polyp segmentation, introducing a reverse attention (RA) module that utilizes background information. However, PraNet-V1 struggles with multi-class segmentation tasks. To address this limitation, we here propose PraNet-V2, which can effectively handle a broader range of tasks, including multi-class segmentation. At the core of PraNet-V2 is our dual-supervised reverse attention (DSRA) module, which incorporates explicit background supervision, independent background modeling, and semantically enriched attention fusion. Our PraNet-V2 framework exhibits strong performance on four polyp segmentation datasets. Moreover, the integration of DSRA into three state-of-the-art semantic segmentation models enables iterative refinement of foreground segmentation, yielding improvements of up to 1.36% in mean Dice score. Jittor code and supplementary materials are available at https://github.com/ai4colonoscopy/PraNet-V2/tree/main/binary_seg/jittor.