We present a novel general speech restoration model, DBP-Net (dual-branch parallel network), designed to effectively handle complex real-world distortions including noise, reverberation, and bandwidth degradation. Unlike prior approaches that rely on a single processing path or separate models for enhancement and restoration, DBP-Net introduces a unified architecture with dual parallel branches-a masking-based branch for distortion suppression and a mapping-based branch for spectrum reconstruction. A key innovation behind DBP-Net lies in the parameter sharing between the two branches and a cross-branch skip fusion, where the output of the masking branch is explicitly fused into the mapping branch. This design enables DBP-Net to simultaneously leverage complementary learning strategies-suppression and generation-within a lightweight framework. Experimental results show that DBP-Net significantly outperforms existing baselines in comprehensive speech restoration tasks while maintaining a compact model size. These findings suggest that DBP-Net offers an effective and scalable solution for unified speech enhancement and restoration in diverse distortion scenarios.