Preheat the oven to 425°F (220°C).
In a small bowl, whisk together the sweet paprika, minced thyme, and salt and pepper to taste (about 1 teaspoon salt and 1/2 teaspoon pepper is suggested).
In a large bowl, combine the potato wedges, cherry tomatoes, sliced poblano chiles, sliced yellow bell pepper, and sliced garlic. Drizzle with 1 tablespoon olive oil and toss to coat evenly.
Spread the vegetables on a rimmed baking sheet, arranging the potatoes cut-side down. Sprinkle about 2 teaspoons of the paprika-thyme seasoning over the vegetables.
Return the mixing bowl to toss the chicken thighs: drizzle with the remaining 2 tablespoons olive oil, toss to coat, then sprinkle with the remaining seasoning so the chicken is evenly seasoned.
Place the seasoned chicken thighs on top of the vegetables on the baking sheet, skin-side up.
Roast in the preheated oven until the chicken is golden and crisp and reaches 180°F (82°C) on an instant-read thermometer, about 45 to 50 minutes.
Remove from the oven and let rest for 5 minutes. Garnish with minced fresh parsley and serve.