In a 5-quart saucepan, warm oil over high heat. Season chicken with salt and pepper; brown both sides, 8 to 10 minutes. Transfer to a plate to cool; slice into thin strips.
Reduce heat to medium. Cook sausage until browned, 1 to 2 minutes. Add onion, bell pepper, and celery; cook until onion is translucent, 4 to 6 minutes. Stir in flour; cook until light brown, stirring constantly, until toasted, 30 seconds to 1 minute.
Stir in tomato paste, thyme, half the okra, tomatoes, and 2 quarts water. Bring to a boil; reduce to a simmer. Stir in chicken and any accumulated juices. Cover; cook until thickened, about 30 minutes.
Stir in remaining okra; simmer, uncovered, until chicken is starting to fall apart, about 30 minutes. Season with salt and pepper. Stir in parsley, if desired.