Recently, several newest Depth Map Super-Resolution (DMSR) methods have incorporated depth edge prediction as auxiliary guidance into the optimization model, creating a dual-task driven unfolding network for enhancing depth edge refinement. Nevertheless, these approaches either overlook explicit dual-edge consistency constraint or merely fuse color edges with depth edges once. This oversight results in diminished generalization ability and subpar performance. To this end, we transform the DMSR problem as a triple-task optimization model explicitly constrained by dual-edge consistency. According to the Alternating Direction Method of Multipliers (ADMM) theory, the proposed model can be cast as iterative sub-optimizations for color-edge update, depth-edge update, depth map update, and augmented Lagrange multiplier update. These sub-optimizations can be further unfolded into an interpretable ADMM network. Within this network, we integrate learnable modules into the initial pure formula expansion, enabling high-throughput information transmission and thereby enhancing the network’s representational power. A large number of experiments have demonstrated that the proposed method achieves better reconstruction results as compared with several DMSR methods.