In a large pot of salted boiling water, cook the pasta according to package instructions until al dente. Drain and set aside, reserving a bit of the pasta water.
In a large skillet, heat the olive oil over medium heat. Add the sliced chorizo and sauté until it begins to brown, about 5-6 minutes.
Add the chopped onion to the skillet and cook until translucent, about 3-4 minutes. Stir in the minced garlic and cook for an additional minute until fragrant.
Lower the heat and pour in the heavy cream, stirring well. Add the smoked paprika and season with salt and pepper. Allow the mixture to simmer gently for 5 minutes until it thickens slightly.
Add the cooked pasta and spinach leaves to the skillet, tossing to coat everything evenly in the creamy sauce. If the sauce is too thick, add a splash of the reserved pasta water until desired creaminess is achieved.
Stir in the grated Parmesan cheese until melted and combined. Taste and adjust seasoning if necessary.
Transfer the creamy chorizo pasta to serving plates, garnish with fresh parsley, and an extra sprinkle of Parmesan if desired.