Melt butter in heavy large pot over medium-high heat. Add onions and sauté until golden, about 15 minutes. Add carrots and parsnips and cook 10 minutes. Add 2 cans of broth, potatoes, parsley and thyme. Cover and simmer until potatoes are tender, stirring occasionally, about 30 minutes.
Puree half of soup in processor. Mix puree into remaining soup in pot. Stir in milk and Sherry. Season to taste with salt and pepper. (Can be prepared 1 day ahead. Cover and refrigerate.)
Bring soup to simmer, thinning with additional broth, if desired. Ladle into bowls and serve.