Coarsely chop the broccoli florets and tender stems and toss with 1 tablespoon of the olive oil, 1/2 teaspoon salt and 1/4 teaspoon pepper. Spread in a single layer on a rimmed baking sheet and roast until tender with the edges crispy, 20 minutes.
Meanwhile, bring a large pot of water to a boil. Add a generous pinch of salt and the pasta. Cook according to the package directions. Reserve 1/2 cup pasta water and drain the pasta. Toss the hot pasta with the roasted broccoli, sun-dried tomatoes, 1/4 cup pasta water, the remaining 1 tablespoon olive oil and the reserved sun-dried tomato oil. Stir in the mozzarella, parsley and as much remaining pasta water as needed to make a light sauce. Season to taste with salt and pepper.