Product codes are considered a promising coding scheme for future optical and wireless communications. However, existing soft-input soft-output (SISO) iterative decoding for product codes involves sorting, list management, and soft-output generation, which includes hardware-unfriendly operations and dataflow. In this paper, we apply belief propagation (BP) decoding to product codes, which leverages the sparse factor graphs of the component codes. As an inherently SISO approach, the proposed product BP decoder can directly propagate soft messages across iterations with a hardware-friendly parallel structure, which outperforms the traditional Chase-Pyndiah decoding with a list size of 64 for various code rates.