In a large pot of boiling salted water, cook pasta according to package instructions. Reserve 1 cup pasta water, then drain pasta and return to pot.
In a large skillet, heat oil over medium-high. Add sausage and cook, breaking meat up with a wooden spoon, until browned, 3 minutes. With a slotted spoon, transfer sausage to pot. Add parsnips to skillet and saute until softened and browned, 5 minutes. Add chard and cook until wilted, 2 minutes. Transfer mixture to pot and toss. Add Parmesan and enough pasta water to create a light sauce that coats pasta; season with salt and pepper. Serve with additional Parmesan.