Marinate the chicken: Combine the olive oil, vinegar, garlic, paprika, and salt in a bowl. Add the chicken pieces and toss to coat. Set aside to marinate while you prepare the vegetables (or marinate in the fridge for up to 1 day).
Assemble the chicken and vegetables: Preheat the oven to 400 degrees F and line a half sheet pan or 2 quarter sheet pants with foil and coat with 1 tablespoon olive oil.
Peel the potatoes if you wish and cut into approximately 3/4-inch chunks. Cut the cauliflower into about 1-inch florets. Cut the onion half into eight thin wedges. Spread the vegetables on the prepared sheet pan and, using your hands, toss with 2 tablespoons olive oil, the thyme, and plenty of salt and pepper. Scatter the olives over the vegetables and the chorizo, if using.
Make spaces throughout the pan of vegetables and add the chicken pieces, skin-side up.
Bake for 30 minutes, at which point check the tray and toss the potatoes and cauliflower to ensure they're cooking evenly, then return to the oven to bake for another 10 to 20 minutes, or until the chicken and vegetables are cooked through
To finish: Toss the pepper strips together with the vinegar, salt, and sugar, and set aside to marinate until needed.
When the chicken and vegetables are ready, top with the pepper strips and chopped parsley.