Heat 1 teaspoon of the oil in a skillet over medium-high heat. Add the kielbasa and cook, turning occasionally, until browned, 2 to 3 minutes. Transfer to a plate.
Add the remaining 2 tablespoons of oil to the skillet and reduce heat to medium. Add the onion and cook, tossing occasionally, until tender, 8 to 10 minutes.
Add the coleslaw mix, caraway seeds, ¾ teaspoon salt, and ½ teaspoon pepper and cook, tossing frequently, 2 minutes more.
Stir in the wine and vinegar. Nestle the kielbasa in the vegetables. Simmer, stirring occasionally, until vegetables are tender and wine has nearly evaporated, 5 to 6 minutes.
Stir in the parsley. Serve with the bread and mustard, if desired.