Remove the sausage from the casings and cook it in a large frying pan over medium heat, breaking it up with a spatula, until no pink remains and the sausage begins to brown. Remove the sausage and set aside, leaving behind the fat.
Bring a large saucepan of water to a boil and salt generously; add the pasta and cook until it is al dente, about 8–10 minutes. Drain the pasta, reserving one cup of the cooking water.
Cook the garlic briefly in the sausage fat, about 30 seconds, then add the collards, olive oil, and red pepper flakes, season with salt and pepper, and stir to coat the collards with the fat. (You may need an additional tablespoon or two of olive oil, depending on how much fat the sausage has rendered.) Cover the pan and cook the collards until tender, about 10 minutes, adding half a cup of water, plus more if needed, to steam the collards.
Add the sausage and pasta to the collards, mixing thoroughly. Add the reserved pasta water, if needed, to moisten the dish. Cook for about 3 minutes to warm the dish through. Stir in the Parmesan.
Serve with additional Parmesan.