Heat the olive oil in a large, deep skillet over medium-high heat. Add the sliced sausage in a single layer and cook until browned on both sides, about 4 to 5 minutes. Transfer the sausage to a plate.
Reduce the heat to medium. Add the onion and apple to the same skillet. Cook, stirring often, until the onion softens and the apple begins to turn tender, about 4 minutes.
Stir in the caraway seeds and paprika and cook for 30 seconds, just until fragrant.
Add the drained sauerkraut, broth or apple juice, mustard, optional brown sugar, and black pepper. Stir well, scraping up any browned bits from the bottom of the skillet.
Return the sausage and any juices to the skillet. Cover and simmer over medium-low heat until the sauerkraut is hot and the flavors have blended, about 5 to 6 minutes.
Taste before adding any salt; sauerkraut and sausage are often salty enough on their own. Sprinkle with parsley if using, then serve hot.