Whisk together the marinade ingredients (olive oil, thyme, chili powder, smoked paprika, and minced garlic) in a bowl large enough to fit the shrimp.
Add the shrimp to the marinade and gently toss to combine. Cover with plastic and set aside in the fridge for 10-15 minutes (or up to 8-9 hours).
Preheat the oven to 400 degrees. Put the shrimp on a foil-lined baking sheet in a single layer. Sprinkle a few pinches each of kosher salt and black pepper over the shrimp.
Roast in the oven for 8-10 minutes tops – the shrimp should be firm. Squeeze some lemon onto the shrimp and top with chopped parsley. Serve immediately with extra lemon wedges.