Turn on a multi-functional pressure cooker (such as Instant Pot) and select Saute function. Add butter and oil and allow butter to melt.
Sprinkle 2 teaspoons Cajun seasoning on each side of the chicken breasts. Add chicken to hot Instant Pot and cook until browned, 2 minutes on each side. Transfer chicken breasts to a plate. Add andouille sausage, mushrooms, and garlic and sauté, about 2 minutes. Cancel Saute mode.
Pour in chicken broth, whipping cream, and hot sauce. Stir in penne pasta. Nestle chicken breasts on top of pasta. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 12 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, for about 20 minutes. Release remaining pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid.
Remove chicken breasts to a cutting board and cut into bite-sized pieces.
Stir Parmesan cheese into pasta sauce in the Instant Pot® until melted. Return chicken back to the pot and stir to combine. Add bell pepper and stir to combine. Residual heat will cook the peppers. Serve immediately garnished with parsley.