Add oil (2 tablespoons) to a large pot heating over medium heat. Saute onion (1 large), garlic (4 large), carrots (6), celery (4 stalks), 1/2 teaspoon salt, black pepper, and Italian seasoning (2 teaspoons) until onion is translucent.
Sprinkle flour (2 tablespoons) over vegetables and cook, stirring constantly, for 1 minute.
Remove the pot from the heat and slowly whisk in chicken stock (8 cups) until smooth.
Stir in Better Than Bouillon (2 tablespoons), lemon juice (1/2 cup), bay leaf (1), rosemary sprig (1), and Parmesan rind and bring to a boil over medium-high heat. Immediately reduce heat and simmer for 30 minutes. Remove the bay leaf and Parmesan rind and discard.
Bring the soup back to a high simmer and add the rice (1/2 cup) and Parmesan cook until it's tender about 15-20 minutes. Add the chicken (2 cups) until it is heated through, about 5-8 minutes.
Adjust seasoning, ladle into individual bowls, and sprinkle with parsley. Serve, passing extra Parmesan.