In a large nonstick saute pan, heat 2 teaspoons of the olive oil over medium-high heat until hot but not smoking. Add the onion and mustard and saute until the onion is soft and lightly golden, about 6 minutes. Transfer to a large bowl.
Reduce the heat to medium and add 2 more teaspoons of the olive oil to the pan. Add the carrot and toss and stir constantly until the carrot is tender-crisp, about 3 minutes. Transfer to the bowl with the onion.
Add the remaining 2 teaspoons oil to the pan over medium heat. Add the cabbage and toss and stir constantly until the cabbage just begins to wilt, about 3 minutes. Quickly transfer the cabbage to the bowl with the other vegetables.
Quickly add the vinegar and honey to the pan over medium heat, stirring until combined and bubbly and the honey is dissolved. Pour over the slaw. Add the salt and pepper and toss well. Garnish with the caraway seed and parsley and serve warm.