Combine 4 1/2 cups water, carrots, 4 tablespoons butter, sugar, and coarse salt in heavy large pot. Bring to boil. Reduce heat; cover and simmer until carrots are just tender when pierced with knife, about 10 minutes. Drain. (Can be prepared 3 hours ahead. Let stand at room temperature.)
Melt remaining 6 tablespoons butter in large nonstick skillet over medium-high heat. Add maple syrup and brown sugar and stir until sugar dissolves. Add carrots and cook until heated through, about 5 minutes. Season with salt and pepper.
Transfer carrots to large bowl. Sprinkle with parsley and serve.