Grab a large glass or plastic bowl and combine the olive oil, minced garlic, dried oregano, dried basil, kosher salt, black pepper, and lemon zest. Give it all a good stir until everything is evenly blended. This marinade is light on ingredients but big on flavor, so don't skip any of the seasonings.
Add the peeled and deveined shrimp directly into the bowl with the marinade. Toss everything together until each piece of shrimp is fully coated. Take your time here since even coating means even flavor once it hits the oven.
Cover the bowl with plastic wrap and place it in the refrigerator for 1 to 3 hours. I never go past the 3-hour mark because the acid from the lemon zest will start to "cook" the shrimp on its own, which throws off the texture. A shorter marinate is perfectly fine if you're short on time.
While the shrimp marinates, preheat your oven to 400°F. Getting the oven fully heated before the shrimp goes in ensures they roast quickly and evenly instead of slowly warming up alongside a cold oven.
Once the shrimp is done marinating, sprinkle the freshly grated Parmesan into the bowl and toss again until it clings to the shrimp. I like to be generous here since the cheese forms a light, savory crust once it roasts.
Pour the shrimp into a 9x13 baking dish or spread it across a large baking sheet in a single layer. Roast at 400°F for about 8 to 10 minutes, or until the shrimp turn pink, firm up, and are cooked all the way through.
Pull the shrimp from the oven and finish with a light drizzle of olive oil and a sprinkle of chopped parsley. Serve immediately while everything is warm and the Parmesan is at its best.