Melt butter in a large cast iron pan over medium heat, then add the flour and cook until deep brown, about 3 to 5 minutes.
Add the onion, celery, bell peppers, and garlic, and cook over moderately low heat, stirring frequently, until the onion is translucent, about 5 minutes.
Stir in the thyme, salt, tomatoes, okra, broth, and bay leaf, and bring to a boil over medium-high heat; cover, reduce heat to low, and simmer for 30 minutes.
Add shrimp and hot sauce, and cook for 3 to 5 minutes or just until shrimp turn pink.
Remove and discard bay leaf.
Serve over hot rice and sprinkle with fresh parsley.