Preheat oven to 425°. Sprinkle chicken thighs all over with ½ tsp. Morton kosher salt and ½ tsp. pepper. Coat a large cast-iron skillet with 1 Tbsp. oil. Arrange chicken thighs, skin side down, in pan and place over medium heat. Cook, undisturbed, until skin underneath is browned, 14–16 minutes
Meanwhile, toss potatoes, peperoncini, garlic, oregano, remaining ¼ tp. kosher salt, and ¼ tsp. pepper in a large bowl. Set aside.
Turn chicken over. Add potato mixture to pan, arranging snugly around thighs. Pour peperoncini brine around edges of pan and give pan a shake to evenly distribute. Transfer skillet to oven and roast until chicken is cooked through and potatoes are tender, 40–45 minutes.
Remove skillet from oven and transfer chicken and peperoncini to a plate. Place skillet over high heat and cook potatoes, stirring often, until coated in drippings and glossy, about 4 minutes. Remove from heat and stir in red onion, sliced celery, and parsley.
Place chicken and pepperoncini on top of vegetables and scatter some celery leaves over.