Bring a large pot of salted water to a boil. Add the pasta and cook according to the package instructions to al dente. Reserve 1 cup pasta water, then drain.
Heat the oil in a large skillet over medium-high heat. Add the zucchini and cook until browned and crisp, about 2 minutes. Add the chicken, rosemary, garlic and lemon zest and juice; toss to combine. Season with salt and red pepper flakes. Add the cream and 1/2 cup pasta water. Bring to a simmer and cook until the sauce is slightly thickened and the zucchini is crisp-tender, 2 to 3 minutes.
Add the pasta to the skillet and toss to coat, adding a little more pasta water if it seems dry. Sprinkle with the Asiago cheese and parsley and toss.