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Step 1
Heat the oven to 475°F with a rack in the middle position. In a large bowl, stir together the oil, oregano, 1¼ teaspoons salt and ½ teaspoon pepper. Add the chicken and potatoes; toss, then rub the seasoning mixture into the chicken and potatoes.
Step 2
Place the garlic in the center of a rimmed baking sheet, then arrange the chicken, skin side up, around the garlic; this placement helps prevent the garlic from scorching during roasting. Arrange the lemons, cut sides up, and the potatoes, cut sides down, in an even layer around the chicken. Drizzle any oil mixture remaining in the bowl over the ingredients. Roast until the chicken is golden brown and the thickest part of the breasts (if using) reaches 160°F and the thickest part of the largest thigh (if using) reaches 175°F, about 30 minutes.
Step 3
Transfer the chicken, potatoes and lemon halves to a serving platter, leaving the garlic in the center. Using a fork, mash the garlic to a rough paste. Squeeze 2 of the lemon halves onto the baking sheet, then add 3 tablespoons water and the parsley. Stir to combine, scraping up any browned bits, then taste and season with salt and pepper. Pour the sauce over and around the chicken and potatoes.