Trim all the skin from the underside of the chicken thighs, but leave the skin on top intact. To ensure that all the potatoes fit in the skillet, halve them crosswise to minimize their surface area. Combining the garlic with lemon juice in step 1 makes the garlic taste less harsh, but only if the lemon juice is added immediately after the garlic is minced.
1. Adjust oven rack to upper-middle position and heat oven to 450 degrees. Pat chicken dry with paper towels and sprinkle on both sides with ¾ teaspoon salt and ¼ teaspoon pepper. Toss potatoes with 1½ teaspoons oil and ½ teaspoon salt. Mince whole garlic clove and immediately combine with lemon juice in small bowl; set aside.
2. Heat remaining 1½ teaspoons oil in 12-inch ovensafe skillet over medium-high heat until shimmering. Place chicken, skin side down, in single layer in skillet and cook, without moving it, until chicken has rendered about 2 tablespoons of fat, about 2 minutes. Place potatoes cut side down in chicken fat, arranging so that cut sides are in complete contact with surface of skillet. Sprinkle chicken and potatoes with oregano and thyme. Continue to cook until chicken and potatoes are deeply browned and crisp, 6 to 8 minutes longer, moving chicken and potatoes to ensure even browning and flipping pieces when fully browned. When all pieces have been flipped, tuck halved garlic cloves among chicken and potatoes. Remove skillet from heat and pour wine into skillet (do not pour over chicken or potatoes). Transfer skillet to oven and roast until potatoes are tender when pierced with tip of paring knife and chicken registers 185 to 190 degrees, 15 to 20 minutes.
3. Transfer chicken and potatoes to deep platter, browned sides up. Place skillet over medium heat (handle will be hot) and stir to incorporate any browned bits. Using slotted spoon, transfer garlic cloves to cutting board. Chop coarse, then mash to smooth paste with side of knife. Whisk garlic paste into sauce. Continue to cook until sauce coats back of spoon, 2 to 3 minutes longer. Remove from heat and whisk in reserved lemon juice mixture and 1½ teaspoons parsley. Pour sauce around chicken and potatoes. Sprinkle with remaining 1½ teaspoons parsley and serve.