Heat a splash of oil in a large pot over medium heat.
Add the onion, carrots, and celery. Sauté until softened, about 5 minutes.
Add the chicken backs, garlic, bay leaf, and thyme. Stir briefly.
Pour in the chicken broth and bring to a boil.
Reduce heat and simmer for about 45 minutes.
(Optional) Add noodles or rice and cook an additional 10 minutes until tender.
Remove chicken backs, shred off any meat, and return meat to the pot.
Season with salt and pepper to taste.
Serve hot, sprinkled with fresh parsley!