Preheat the oven to 425°F (220°C). Line a large sheet pan with parchment paper or lightly grease it.
In a large bowl, combine the halved baby potatoes with 2 tablespoons of olive oil, 2 cloves of minced garlic, smoked paprika, dried oregano, salt, and black pepper. Toss until the potatoes are evenly coated.
Spread the potatoes in a single layer on one side of the prepared sheet pan. Roast in the preheated oven for 15 minutes.
While the potatoes roast, in the same large bowl, combine the broccoli florets with 1 tablespoon of olive oil, 2 cloves of minced garlic, lemon zest, and a pinch of salt and pepper. Toss to coat.
After the potatoes have roasted for 15 minutes, remove the sheet pan from the oven. Add the broccoli florets to the other side of the pan in a single layer.
In the same bowl, add the peeled and deveined shrimp. Add lemon juice, red pepper flakes if using, and toss gently to combine.
Spread the shrimp evenly over the broccoli and potatoes on the sheet pan.
Return the sheet pan to the oven and roast for an additional 5 to 7 minutes, or until the shrimp are pink and opaque and the broccoli is tender-crisp.
Remove the sheet pan from the oven and sprinkle the chopped fresh parsley over the shrimp, broccoli, and potatoes.
Serve immediately.