Heat the vegetable oil in a large heavy pot or Dutch oven over medium-high heat until shimmering.
Add the andouille sausage and cook, stirring occasionally, until well browned, 6–8 minutes; transfer to a plate.
Add the chicken thighs to the pot and brown on both sides, 5–7 minutes total; leave the drippings in the pot.
Add the onion, green bell pepper, and celery; cook, scraping up browned bits, until softened, 6–8 minutes.
Stir in the garlic and cook until fragrant, about 30 seconds.
Return the sausage to the pot. Sprinkle in the paprika, dried thyme, dried oregano, cayenne pepper, black pepper, and kosher salt; stir and cook 30 seconds to bloom the spices.
Stir in the long-grain white rice and cook, stirring, 1 minute to lightly toast.
Add the canned diced tomatoes with their juices, the chicken stock, and the bay leaves; bring to a boil, then reduce to low, cover, and simmer undisturbed until the rice is just tender and most liquid is absorbed, 20–25 minutes. Doneness cue: rice should be tender with small steam holes on the surface.
Scatter the raw shrimp over the rice, cover, and cook on low until the shrimp are pink and opaque, 4–6 minutes.
Remove from heat, discard the bay leaves, and let rest covered 5 minutes. Fluff with a fork, garnish with the flat-leaf parsley, and serve with hot sauce.