Cook spaghetti according to package directions until al dente. Drain and set aside.
In a large skillet, heat olive oil over medium heat. Sauté garlic, onion, and bell peppers until soft and fragrant.
Add cooked shredded chicken to the skillet and cook for 2–3 minutes until heated through.
In a separate saucepan, melt butter over medium heat. Whisk in flour and cook for 1 minute until lightly golden.
Gradually pour in chicken broth and milk, whisking constantly until smooth and slightly thickened.
Stir in cheddar and Parmesan cheeses until melted and creamy. Season with salt, pepper, paprika, and oregano.
Combine the cooked spaghetti and chicken-vegetable mixture with the creamy sauce. Toss well to coat evenly.
Serve hot, garnished with chopped parsley if desired.