Cook the penne pasta according to package directions until al dente. Reserve 1 cup of pasta water before draining and set aside.
While the pasta cooks, toss the chicken with 2 tbsp Cajun seasoning and 1 tsp salt. Heat the olive oil in a large skillet over medium-high heat. Add the chicken and cook for 5-7 minutes until browned and cooked through. Transfer to a plate and set aside.
Reduce the heat to medium and add the butter to the same skillet. Add the onion, red bell pepper, and green bell pepper. Cook for 5-6 minutes until softened.
Add the garlic, tomato paste, and remaining 1 tbsp Cajun seasoning. Cook for 1 minute, stirring constantly. Pour in the chicken broth and scrape up any browned bits from the bottom of the pan. Stir in the heavy cream, salt, and black pepper. Bring to a gentle simmer.
Add the Parmesan cheese and stir until melted and smooth. Add reserved pasta water a few tablespoons at a time if needed to reach your desired consistency.
Return the chicken to the skillet and stir to combine.
Add the drained pasta and toss until evenly coated in the sauce. Cook for 1-2 minutes until everything is heated through.
Garnish with fresh parsley and additional Parmesan cheese before serving.
