In a large Dutch oven, heat oil over medium heat. Whisk in flour, and stir constantly until a dark- brown roux forms, about 12 to 14 minutes.
Add onion, celery, bell pepper, and garlic; cook until tender, 4 to 5 minutes.
Add tomato paste, tomato, jalapeno, bay leaf, thyme and hot sauce. Slowly add broth, stirring constantly, and bring to a simmer. Reduce heat to medium-low, and cook, stirring frequently, 30 minutes.
Gently stir in alligator, increase heat to medium, and cook, stirring often, until tender, 30 to 45 minutes. Season to taste with salt and peppers. Additional broth may be added if sauce piquant becomes too thick. Remove and discard bay leaf. Add green onion and parsley, and serve over rice.