Cook the pasta according to package directions until al dente. Drain, reserving about 1/2 cup of the pasta water, and set aside.
While the pasta cooks, heat the olive oil in a large skillet over medium-high heat. If using chicken, season it with salt and pepper and cook until browned and cooked through. Remove the chicken from the skillet and set aside.
Add the chopped onion and bell peppers to the same skillet. Sauté for 5-7 minutes, or until the vegetables are tender-crisp. Add the minced garlic, thyme, allspice, and cayenne pepper. Cook for another minute until fragrant.
Pour in the heavy cream and chicken broth. Bring to a simmer, stirring constantly. Let it simmer for 2-3 minutes until the sauce begins to thicken slightly.
Stir in the grated Parmesan cheese until melted and the sauce is smooth. Season with salt and pepper to taste.
Return the cooked chicken (if using) to the skillet. Add the drained pasta to the sauce. Toss everything together until the pasta and chicken are well coated in the creamy sauce.
If the sauce seems too thick, add a little of the reserved pasta water, a tablespoon at a time, until it reaches your desired consistency.
Stir in the fresh parsley. Serve immediately.