Pat the chicken thighs dry.
Whisk the lemon juice, lemon zest, olive oil, paprika, black pepper, rosemary, thyme, oregano, kosher salt, and red pepper flakes.
Add the chicken and coat well, including under the skin. Cover and refrigerate 2–8 hours, or overnight.
Toss the potatoes, carrots, and garlic with olive oil, salt, pepper, paprika, and oregano.
Stir together all yogurt sauce ingredients. Chill until serving.
Preheat the air fryer to 380°F (193°C).
Air-fry the vegetables 10 minutes.
Shake off excess marinade from the chicken. Add the thighs skin-side up next to the vegetables.
Air fry 18–22 minutes more, flipping the chicken once, until the vegetables are tender and the chicken reaches 165°F (74°C).
Spread the yogurt sauce on a plate. Top with the chicken, potatoes, carrots, and garlic. Finish with chopped parsley.