Whisk together half of the yogurt, lemon juice, olive oil, minced garlic, tomato paste, paprika, and allspice in a large bowl. Season with salt and pepper.
Add the chicken pieces to the marinade, tossing to coat thoroughly. Cover and refrigerate for at least 2 hours.
Preheat the grill to medium-high heat. If using wooden skewers, soak them in water for 30 minutes.
Thread the marinated chicken, cherry tomatoes, yellow squash rounds, and red onion wedges alternately onto the skewers.
Grill the skewers for 10 to 12 minutes, turning occasionally, until the chicken reaches an internal temperature of 165°F and the vegetables are tender with a slight char.
Prepare the yogurt sauce by mixing the remaining yogurt with diced cucumber, parsley, a splash of lemon juice, and a pinch of salt.
Serve the hot chicken skewers immediately with the cucumber yogurt sauce on the side.