Melt the butter in a Dutch oven or large pot over medium heat. Add the onions, carrots, celery, and garlic and cook, stirring occasionally, until the vegetables are soft, about 5 minutes. Add the tomato paste, stir to coat the vegetables, and cook for 1 minute more.

Add the broth, turkey, bay leaves, and thyme, and stir to combine. Bring to a boil, then reduce the heat and simmer for 30 minutes. Stir once more and ladle the soup into bowls. Top with the parsley and serve.