Bring a large pot of salted water to a boil. Add the pasta and cook according to package instructions until al dente. In the last 3 minutes of cooking, add the broccoli florets to the pot. Drain the pasta and broccoli, reserving 1/2 cup of the pasta water.

In the same pot, heat the olive oil over medium heat. Add the minced garlic and red pepper flakes (if using) and sauté for about 1 minute, until fragrant.

Return the drained pasta and broccoli to the pot. Add the reserved pasta water, salt, black pepper, Parmesan cheese, and lemon juice. Toss everything together until well combined and heated through.

Serve immediately, garnished with fresh parsley and additional Parmesan cheese if desired.