Step 1Heat oil in 3-quart saucepan over medium-high heat. Add onion and cook until softened, stirring often, about 4 minutes. Add celery root, parsley root, potato, broth, nutmeg, and salt and pepper to taste. Bring to boil, reduce heat and simmer, partially covered, until vegetables are tender (check parsley root), 25 to 35 minutes.
Step 2Strain vegetables from broth, reserving both; puree vegetables in batches in blender, adding 1 to 1 1/2 cups broth to make smooth soup consistency. Return to pot. Taste and adjust seasonings. Serve hot, garnished with parsley, if using. (Can be made ahead and refrigerated up to 3 days or frozen up to 1 month. Gently reheat, adding water if too thick, and adjust seasoning.)