In a large skillet over medium heat, add the ground beef and pork sausage. Cook until browned, breaking the meat apart with a wooden spoon, for about 5-7 minutes. Drain excess fat if necessary.
Add the diced onion, bell pepper, celery, and minced garlic to the skillet. Sauté until the vegetables are softened, about 5 minutes.
Stir in the rice, chicken broth, Cajun seasoning, thyme, salt, black pepper, and cayenne pepper (if using). Bring to a boil.
Once boiling, reduce the heat to low, cover, and simmer for 20-25 minutes, or until the rice is cooked and has absorbed the liquid.
Remove from heat and let sit covered for an additional 5 minutes. Fluff the rice with a fork.
Serve hot, garnished with sliced green onions and chopped parsley.