Bring a large pot of salted water to a boil. Add the pasta and cook according to the package directions for al dente, adding the broccolini for the last 4 minutes of cooking. Reserve 2 1/2 cups of the cooking water, then drain the pasta and broccolini and set aside.
Meanwhile, heat a large Dutch oven over medium heat and add the olive oil. When the oil is hot, add the sausage and cook, breaking it up with a wooden spoon, until browned, about 4 minutes. Add the onion and cook, stirring occasionally, until softened, about 5 minutes.
Add the tomato paste, red pepper flakes and garlic to the pot and cook, stirring, until sizzling, 30 seconds. Add 2 cups of the reserved pasta water, bring to a simmer and cook until reduced by half, 5 to 6 minutes.
Add the pasta, broccolini and parsley to the sauce and toss to coat, adding more water if needed. Taste and add salt if necessary. Remove from the heat and sprinkle with the grated cheese, then toss well. Top each serving with more grated cheese.