Bring a large pot of well-salted water to a boil over high heat. Add the penne and cook according to package instructions until al dente.

Meanwhile, heat the oil in a 12-inch skillet over medium-high heat. Add the zucchini and cook, stirring occasionally, until barely tender and just starting to brown, about 3 minutes. Add the garlic and cook, stirring occasionally, until the garlic is soft and fragrant, an additional 2 to 3 minutes. Remove from the heat. Stir in the lemon zest, 3/4 teaspoon salt, and 1/4 teaspoon pepper.

Reserve 1/2 cup pasta cooking water and drain the pasta. Add the pasta to the zucchini mixture along with the Parmigiano, mint, basil, and parsley. Season to taste with salt and pepper. Toss to combine, adding some of the reserved water if necessary to moisten the pasta, and serve immediately.