Chop broccoli rabe into 1-inch pieces and set aside.
Place a large skillet over medium heat. When hot, add the sausage and oregano and cook, breaking up with a wooden spoon 6 to 8 minutes, until browned. Transfer to a plate.
Add the broccoli rabe and parsnip noodles to the skillet with the garlic and red pepper flakes, then add broth and cook until the noodles are al dente, about 5 minutes.
Return sausage to the skillet, add the cheese and stir to combine.