Sparse neural networks can greatly facilitate the deployment of neural networks on resource-constrained platforms as they offer compact model sizes while retaining inference accuracy. Because of the sparsity in parameter matrices, sparse neural networks can, in principle, be exploited in accelerator architectures for improved energy-efficiency and latency. However, to realize these improvements in practice, there is a need to explore sparsity-aware hardware-software co-design. In this paper, we propose a novel silicon photonics-based sparse neural network inference accelerator called SONIC. Our experimental analysis shows that SONIC can achieve up to 5.8x better performance-per-watt and 8.4x lower energy-per-bit than state-of-the-art sparse electronic neural network accelerators; and up to 13.8x better performance-per-watt and 27.6x lower energy-per-bit than the best known photonic neural network accelerators.