Preheat oven to 400°. Process 3 garlic cloves in food processor until finely chopped. Add 1/2 cup oil, piquillo peppers, honey, 1/2 teaspoon paprika, vinegar, orange juice, and 1/2 teaspoon each sea salt and black pepper; pulse until chunky, and set aside.
Place shallots, lemons, orange, and remaining 10 garlic cloves on a large, rimmed baking sheet (or divide among 2 jelly-roll pans). Drizzle with 3 tablespoons olive oil; sprinkle with 1/4 teaspoon paprika and 1/2 teaspoon each sea salt and black pepper.
Heat 3 tablespoons oil in a large skillet over medium-high heat until hot. Dredge chicken in flour, shaking off excess. Sprinkle with 1/4 teaspoon each sea salt and black pepper. Cook chicken, in batches, adding more oil if necessary; cook 3 minutes on each side or until golden brown. Transfer to baking sheet. Drizzle with 2 tablespoons oil; sprinkle with remaining 1/4 teaspoon each paprika, sea salt, and black pepper. Bake 30 to 35 minutes or until chicken is done. Transfer chicken to a platter with garlic, shallots, lemons, and orange; top with parsley.
Tip: The high heat and bold flavors in the sauce will blunt the delicate flavor of extra virgin olive oil, so use a good-quality bulk variety.