Cook the pasta according to the package directions. Reserve ½ cup of the pasta water, drain, and return the pasta to the pot.
Meanwhile, heat the oil in a large skillet over medium heat. Add the zucchini and ½ teaspoon salt.
Cook the zucchini, tossing often, until just tender, 4 to 5 minutes.
Add the chickpeas, garlic, and red pepper and cook until heated through, 2 to 3 minutes.
Toss the pasta with the reserved pasta water and ¼ cup of the Parmesan.
Divide the pasta among bowls and top with the zucchini mixture and the remaining ¼ cup of Parmesan.