Place the potatoes in a microwave-safe bowl with 2 tablespoons water (30 mL). Cover loosely and microwave on high until nearly tender, about 5 to 7 minutes. Drain any water and set aside. Alternatively, simmer the potatoes in water until nearly tender, then drain.
Heat the olive oil in a large, deep skillet over medium-high heat. Add the kielbasa in a single layer and cook until browned on both sides, about 4 to 5 minutes. Transfer the kielbasa to a plate.
Reduce the heat to medium. Add the onion to the skillet and cook, stirring often, until softened and lightly golden, about 5 minutes. Add the apple and cook for 2 minutes more.
Stir in the drained sauerkraut, broth, mustard, brown sugar, caraway seeds if using, and black pepper. Scrape up any browned bits from the bottom of the skillet.
Add the nearly tender potatoes and browned kielbasa. Stir gently to combine. Cover and cook over medium-low heat until the potatoes are tender and the mixture is hot throughout, about 8 to 10 minutes.
Taste before adding any salt, since kielbasa and sauerkraut can be salty. Sprinkle with parsley, if desired, and serve hot.