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 about 1/2 cup pasta water, then drain.
Toss the broccoli florets with 1 tablespoon olive oil, 1/2 teaspoon salt, and 1/2 teaspoon black pepper on a baking sheet. Roast for 15 minutes, until tender and slightly browned.
While broccoli roasts, heat a large skillet over medium-low heat and add 5 tablespoons olive oil. Add the sliced garlic and crushed red pepper flakes and cook, stirring occasionally, until the garlic is just golden—be careful not to burn it—then reduce heat to very low or turn off.
Remove the baking sheet from the oven and push the roasted broccoli to one side. Arrange the shrimp in a single layer on the empty side, drizzle with the remaining 1 tablespoon olive oil, and season with salt and pepper. Roast for 6–8 minutes, until shrimp are opaque and pink.
Add the drained bucatini to the skillet with the garlic oil and toss to coat, adding a few tablespoons of reserved pasta water if needed to loosen the sauce.
Add the roasted shrimp and broccoli to the pasta, toss in the chopped parsley, and adjust seasoning. Squeeze lemon over the dish to taste.
Serve immediately with grated Parmesan and extra crushed red pepper if desired.