Bring a large pot of salted water to a boil. Cook the pasta according to the package instructions until al dente. Reserve about ½ cup of the pasta water, then drain.
Heat the olive oil and butter in a large skillet over medium heat.
Add the chopped onion and cook for 3–4 minutes until softened.
Add the minced garlic and cook for about 30 seconds until fragrant.
Add the shrimp, squid, mussels, and fish. Cook for 2–3 minutes, turning gently, until the seafood begins to cook.
Stir in the diced tomatoes, seafood stock, paprika, oregano, black pepper, salt, and red chili flakes if using.
Simmer for about 4–5 minutes until the seafood is cooked through and the sauce begins to reduce.
Lower the heat and stir in the heavy cream and Parmesan cheese.
Simmer gently for 2–3 minutes until the sauce becomes creamy. Do not let it boil rapidly.
Add the cooked pasta to the skillet and toss gently until evenly coated with the sauce.
Add a little reserved pasta water if the sauce is too thick.
Stir in the fresh lemon juice and taste for seasoning.
Garnish with chopped parsley and extra Parmesan cheese, if desired.
Serve immediately with lemon wedges.