Prepare the Parsley: Separate the tough stems from the delicate leaves and tender stems of the parsley. Roughly chop the tough stems for the marinade.

Make the Marinade: In a food processor, combine the chopped parsley stems, garlic, jalapeño, capers, lemon juice, sugar, salt, and pepper along with the buttermilk. Blend until smooth. Transfer the mixture to a medium bowl and add the chicken thighs. Toss to coat evenly and cover. Refrigerate for a minimum of 30 minutes or up to 24 hours to marinate.

Cook the Parsley: Heat a large skillet, preferably nonstick, over medium-high heat and add olive oil. Add the reserved parsley leaves and tender stems, cooking them while stirring occasionally, until wilted, bright green, and slightly charred in spots. This should take about 2 to 3 minutes. Season with salt and pepper, then transfer the cooked parsley to a serving platter.

Cook the Chicken: In the same skillet, add enough olive oil to coat the bottom. Add the marinated chicken thighs, discarding any excess marinade. Sear the chicken for 5 to 7 minutes per side, or until the exterior is browned and caramelized. Once done, transfer the chicken to the platter with the parsley. Serve with lemon wedges on the side for spritzing.