Melt 1 1/2 tablespoons butter in heavy large skillet over medium-high heat. Add onions and cook until golden, stirring often, about 15 minutes. Mix in rosemary. (Can be made 4 hours ahead. Cover and let stand at room temperature.)

Bring large pot of salted water to boil. Add parsnips and potatoes; boil until tender, about 20 minutes. Drain well. Return vegetables to pot; mash. Add warm milk and 4 1/2 tablespoons butter; stir until smooth. Stir in onion mixture. Season with salt and pepper. (Can be made 6 hours ahead. Cover; chill. Rewarm over medium-high heat, stirring frequently and adding more milk if too dry.)