1. Heat oil or margarine in large pot over medium heat. Add onion; Sauté until soft, about five minutes.
2. Add parsnips, potatoes, and turnips; stir five minutes. Add garlic; stir one minute. Add chicken soup; bring to a boil.
3. Cover; reduce heat to medium-low. Simmer until vegetables are tender, about 15–20 minutes.
4. Using slotted spoon, transfer vegetables to large bowl; reserve soup. Add mayonnaise to vegetables.
5. Using a potato masher, mash vegetables to a smooth puree. Add two tablespoons soup to pureed vegetables. You may add additional soup, to reach desired creamy consistency.
6. Season with salt and pepper.