Cook the pasta according to package instructions until al dente.
In a large skillet, heat olive oil over medium heat.
Add the chorizo to the skillet and cook for about 5-7 minutes, breaking it apart with a spoon.
Add minced garlic and red pepper flakes, cooking for an additional 1-2 minutes until fragrant.
Stir in the broccoli florets and a splash of Reserved pasta water, cooking for another 3-4 minutes until the broccoli is tender.
Combine the cooked pasta with the chorizo and broccoli mixture, tossing to coat everything evenly. If needed, add more pasta water.
Season with salt, pepper, and sprinkle with Parmesan cheese before serving.