Peel the carrots and cook them in boiling water until they are partially cooked but still crisp, 4 to 6 minutes. Drain the carrots and rinse them under cold water for at least 10 seconds to stop the cooking. Drain well and pat dry with paper towels.
Prepare the grill for cooking over high heat (500°F).
Lay the carrots flat on a work surface. In a small saucepan over medium heat on the stove, melt the butter with the vinegar and nutmeg. Brush the carrots with about half the butter mixture and season with half the salt and pepper.
Brush the cooking grates clean. Grill the carrots over high heat, with the lid open, until lightly charred with spots and stripes, 3 to 5 minutes, turning occasionally. Transfer the carrots to a platter, brush them with the remaining butter mixture, and season them with the remaining salt and pepper. Sprinkle the parsley over the top, if using. Serve warm.