1. Bring a pot of salted water to boil. Add broccoli; cook until just tender, 3 minutes. Scoop out broccoli with a slotted spoon; transfer to a bowl. Add pasta and cook until tender, about 10 minutes.
2. Warm oil in a large skillet over medium-high heat. Add garlic and crushed red pepper; sauté until fragrant, about 1 minute. Add chickpeas, reserved broccoli and 1/2 tsp. salt and cook, stirring, until heated through.
3. Drain pasta, reserving 1/2 cup of cooking liquid. Return pasta to pot and stir in chickpea and broccoli mixture and lemon juice. Stir in reserved cooking liquid, 1 Tbsp. at a time, until pasta is lightly moistened. Spoon pasta into bowls, sprinkle with Parmesan and serve.