Bring a large pot of generously salted water to a boil.
Add the pasta and cook according to the package directions.
During the last 3–4 minutes of cooking, add the broccoli florets to the same pot.
Reserve ¼ cup of the pasta water, then drain the pasta and broccoli together.
Heat the olive oil in a large skillet over medium heat.
Add the sliced garlic and cook for 30–60 seconds until fragrant.
Stir in the red pepper flakes, if using.
Add the cooked pasta and broccoli to the skillet.
Pour in the reserved pasta water and toss until everything is evenly coated.
Stir in the Parmesan cheese, salt, black pepper, and lemon juice, if using.
Remove from the heat and let the pasta rest for 5 minutes so the sauce coats the pasta evenly.
Garnish with chopped parsley and extra Parmesan before serving.