Heat olive oil in a large heavy-bottomed pot or Dutch oven over medium-high heat.
Add sliced andouille sausage and cook for 4-5 minutes until browned. Remove sausage and set aside.
Add chicken pieces to the pot and cook for 5-6 minutes until browned on all sides. Remove and set aside with sausage.
Reduce heat to medium. Add diced onion, green bell pepper, and celery to the pot. Cook for 5 minutes until vegetables soften.
Add minced garlic and cook for 1 minute until fragrant.
Stir in diced tomatoes with juice, smoked paprika, dried thyme, dried oregano, cayenne pepper, black pepper, salt, and crushed red pepper flakes. Cook for 2 minutes to combine flavors.
Pour in chicken broth and bring to a boil.
Add uncooked penne pasta to the pot. Stir well to submerge pasta in liquid.
Return sausage and chicken to the pot. Reduce heat to medium-low, cover, and simmer for 15 minutes, stirring occasionally to prevent sticking.
After 15 minutes, add raw shrimp to the pot. Cover and cook for an additional 5 minutes until shrimp are pink and cooked through and pasta is tender.
Remove pot from heat. Stir in sliced green onions and chopped fresh parsley.
Serve hot, garnished with additional parsley if desired.