Set your Instant Pot to sauté mode and add olive oil. Once hot, add the sliced sausage and cook for 3-4 minutes until browned. Remove and set aside.
Add chicken pieces to the pot and cook for 4-5 minutes until lightly browned (they don't need to be fully cooked). Remove and set aside with the sausage.
Add onion, bell pepper, and celery to the pot. Sauté for 4-5 minutes until softened. Add garlic and cook for 1 more minute.
Press cancel. Add diced tomatoes, rice, chicken broth, Cajun seasoning, thyme, paprika, and bay leaves. Stir well, scraping up any browned bits from the bottom of the pot.
Return the sausage and chicken to the pot and nestle them into the rice mixture. Do not stir at this point—just press them down gently.
Close the lid, set valve to sealing, and cook on high pressure for 8 minutes.
When done, let the pressure release naturally for 10 minutes, then quick release any remaining pressure.
Remove bay leaves. Gently stir in the raw shrimp, close the lid (no pressure), and let sit for 5 minutes. The residual heat will cook the shrimp perfectly!
Fluff with a fork and garnish with green onions and fresh parsley. Serve with hot sauce on the side for those who like it spicy!