Halve onion lengthwise and cut crosswise into 1/4-inch thick slices. In a large saucepan cook onion in water, covered, over moderate heat, stirring occasionally, 5 minutes, or until softened. Remove cover and cook onion until any liquid in saucepan is evaporated. Preheat oven to 425 degrees.
Peel parsnips and potatoes. Grate parsnips coarse and cut potatoes crosswise into 1/8-inch thick slices. In a 2 cup measure or bowl whisk together milk, broth, and flour until combined well. Add milk mixture to onion with parsnips, potatoes, nutmeg, and salt to taste and simmer, stirring frequently, 1 minute.
Pour vegetable mixture carefully into a shallow 1 1/2-quart baking dish and bake in lower third of oven 45 minutes, or until top is golden brown and potatoes are tender when pierced with a knife.