Improving the quality of underwater images is essential for advancing marine research and technology. This work introduces a sparsity-driven interpretable neural network (SINET) for the underwater image enhancement (UIE) task. Unlike pure deep learning methods, our network architecture is based on a novel channel-specific convolutional sparse coding (CCSC) model, ensuring good interpretability of the underlying image enhancement process. The key feature of SINET is that it estimates the salient features from the three color channels using three sparse feature estimation blocks (SFEBs). The architecture of SFEB is designed by unrolling an iterative algorithm for solving the $\ell_1$ regulaized convolutional sparse coding (CSC) problem. Our experiments show that SINET surpasses state-of-the-art PSNR value by $1.05$ dB with $3873$ times lower computational complexity.