Preheat a grill to medium-high heat (about 400°F). Halve the cherry tomatoes, drain and quarter the artichokes, slice the red onion, chop the sun-dried tomatoes, and halve the olives. Cut the potatoes into 1/2-inch cubes. Tear 12 sheets of heavy-duty foil (roughly 12x18 inches) to make 6 double-layer packets.
In a large bowl, whisk 2 tablespoons olive oil with the garlic, oregano, thyme, paprika, lemon zest, salt, and pepper. Add the chicken thighs and toss to coat. Let stand 10 minutes while you finish the vegetables.
In a separate bowl, toss the potatoes with the remaining 1 tablespoon olive oil and a pinch of salt.
Build the packets: On each double layer of foil, mound a portion of potatoes in the center. Top with a marinated chicken thigh, then scatter tomatoes, artichokes, olives, sun-dried tomatoes, and red onion around the chicken. Fold the foil up and crimp to seal tightly, leaving a little headspace inside for steam.
Grill the sealed packets seam-side up for 22 to 25 minutes, rotating once halfway through, until the potatoes are beginning to soften and the chicken juices run mostly clear.
Open each packet carefully (watch the steam) and fold the edges down to create a shallow tray. Drizzle the vinegar evenly among the packets and sprinkle the feta over the top. Return to the grill, uncovered, for 8 to 12 minutes, until the chicken reaches 165°F and the edges of the vegetables are lightly caramelized.
Finish with parsley and a squeeze of lemon over each packet. Serve hot, right in the foil for easy cleanup.