Computer Vision (CV)-based continuous, automated and precise salmon welfare monitoring is a key step toward reduced salmon mortality and improved salmon welfare in industrial aquaculture net pens. Available CV methods for determining welfare indicators focus on single indicators and rely on object detectors and trackers from other application areas to aid their welfare indicator calculation algorithm. This comes with a high resource demand for real-world applications, since each indicator must be calculated separately. In addition, the methods are vulnerable to difficulties in underwater salmon scenes, such as object occlusion, similar object appearance, and similar object motion. To address these challenges, we propose a flexible tracking framework that uses a pose estimation network to extract bounding boxes around salmon and their corresponding body parts, and exploits information about the body parts, through specialized modules, to tackle challenges specific to underwater salmon scenes. Subsequently, the high-detail body part tracks are employed to calculate welfare indicators. We construct two novel datasets assessing two salmon tracking challenges: salmon ID transfers in crowded scenes and salmon ID switches during turning. Our method outperforms the current state-of-the-art pedestrian tracker, BoostTrack, for both salmon tracking challenges. Additionally, we create a dataset for calculating salmon tail beat wavelength, demonstrating that our body part tracking method is well-suited for automated welfare monitoring based on tail beat analysis. Datasets and code are available at https://github.com/espenbh/BoostCompTrack.