Preheat the oven to 425°F (220°C). Bring a large pot of salted water to a boil and cook the bucatini according to package directions until al dente; reserve 1/2 cup pasta water and drain.
Toss the broccoli florets with 1 tablespoon olive oil, 1/2 teaspoon salt, and 1/2 teaspoon freshly cracked black pepper on a baking sheet; roast for 15 minutes.
While the broccoli roasts, heat a large skillet over medium-low and add 5 tablespoons olive oil. Add the thinly sliced garlic and crushed red pepper flakes and cook, stirring occasionally, until the garlic is just golden, about 2–3 minutes. Keep the heat low or remove from heat to avoid burning.
Remove the baking sheet, push the roasted broccoli to one side, place the shrimp in a single layer on the other side, drizzle with the remaining 1 tablespoon olive oil, and season with salt and pepper. Roast the shrimp 6–8 minutes, until opaque and pink.
Add the drained bucatini to the skillet with the garlic oil and toss to coat, adding a splash of reserved pasta water if needed to loosen the sauce.
Toss in the roasted shrimp and broccoli, add the chopped parsley, and gently combine. Adjust seasoning if needed and spritz with lemon juice from the wedges.
Serve immediately with grated Parmesan and extra crushed red pepper flakes, if desired.