Pat the chicken drumsticks completely dry. Coat with olive oil and season with Cajun seasoning, smoked paprika, garlic powder, onion powder, thyme, black pepper, and salt as needed. Cook the white rice separately according to its package directions while the chicken cooks.
Heat a large deep skillet or Dutch oven over medium-high heat. Sear the drumsticks for 3 to 4 minutes per side until browned. Transfer to a plate.
Reduce the heat to medium. Add the vegetable oil and flour. Cook for 3 to 4 minutes, stirring continuously, until the roux reaches a medium caramel-brown color.
Add the onion, bell pepper, and celery. Cook for 5 to 7 minutes. Add garlic and cook for 30 seconds.
Gradually pour in the chicken broth while whisking. Add Worcestershire sauce, Cajun seasoning, smoked paprika, thyme, and optional cayenne.
Bring the gravy to a gentle simmer and cook for 3 to 5 minutes.
Return the drumsticks to the skillet and spoon gravy and vegetables over the chicken.
Cover, reduce the heat to medium-low, and simmer for 30 to 35 minutes, turning once.
Check the thickest part of the chicken without touching the bone. It must reach at least 165°F.
Remove the lid and simmer for 5 to 10 minutes if the gravy needs to thicken.
Stir in the butter and the 2 tablespoons of chopped fresh parsley, taste, and adjust the seasoning.
Serve the chicken and gravy over white rice. Finish with parsley or green onions.