In a large saucepan melt 3 tablespoons butter over medium heat. Add parsnips and season with 1/2 teaspoon salt. Reduce heat to low and cover. Cook parsnips for 35 to 45 minutes or until very tender, stirring often to prevent any browning. In a separate saucepan repeat with 3 tablespoons remaining butter, carrots, potatoes, leek, and salt.
Using an immersion blender or food processor, puree parsnips with enough cream (about 3/4 cup) and 1 tablespoon remaining butter to make smooth and fluffy. Repeat with carrot-potato mixture, 1/2 cup cream, and remaining butter. Add additional cream to each mixture as needed. Season each mixture with nutmeg. To serve swirl mixtures together in a serving dish. Sprinkle with snipped fresh chives.