Grate the lemon zest into the yogurt and stir in a crushed garlic clove and a pinch of salt; set aside.
In a bowl or large, heavy zip-lock bag, combine the chicken thighs, olive oil, the remaining garlic cloves, the lemon juice, green onions, thyme and a big pinch of salt. Refrigerate for at least an hour, or overnight.
Cover the potatoes with water in a saucepan, add a big pinch of salt, bring to a simmer and cook for 10 minutes, or until just tender.
Heat a large, heavy skillet over medium-high heat, add a generous drizzle of oil and cook the thighs, opened up so they lay flat, until golden on both sides and cooked through.
Transfer to a plate and add the potatoes to the pan; cook until golden on all sides, seasoning with salt.
Serve with the chicken, drizzled with the yogurt and topped with parsley.