
Scatter the carrots in a large skillet and add 1 teaspoon salt and 1 cup water. Bring to a rapid simmer over medium heat and cook until the carrots are just tender, about 5 minutes.
Add the brown sugar, butter, vinegar, remaining 1/4 teaspoon salt and a few grinds of pepper to the skillet. Increase the heat and boil, tossing the carrots occasionally, until the sauce is reduced to a glaze, 4 to 5 minutes. Sprinkle with the parsley and season with salt and pepper.