Peel the carrots and slice them into approximately ¼-inch coins.
Add the carrots, butter, brown sugar, salt, pepper, and 3 tablespoons water to a large skillet.
Cover and cook over medium heat for 8–10 minutes, stirring occasionally, until the carrots are beginning to become tender. If the pan becomes dry before the carrots are tender, add another tablespoon of water.
Remove the lid and continue cooking for 3–5 minutes, stirring occasionally, until the carrots are tender and the liquid has reduced to a buttery glaze.
Remove the skillet from the heat and stir in the vanilla extract.
Sprinkle with chopped parsley, if desired, and serve warm.