In a Dutch over or soup pot, combine parsnips, carrots, onions, sweet potato, turnip, celery, bay leaves and broth.
Bring to boil, then reduce heat and simmer, covered, 30 minutes or until tender.
Remove from heat, remove bay leaves and let cool 20 minutes.
Transfer in small batches to a blender or food processor, cover and puree until smooth. Return to pot, add half-and-half, tarragon and pepper, and heat through.