Toss chicken with cumin, kosher salt, and pepper in a medium bowl. Heat 1 Tbsp. oil in a large skillet over high. Add chicken mixture and cook, stirring occasionally, until lightly browned and cooked through, about 5 minutes. Transfer to a serving bowl. Meanwhile, heat broiler. Divide cucumbers and tomato among small serving bowls. Spoon hummus and yogurt into separate serving bowls. Broil pitas just until warmed, about 2 minutes. Serve chicken, cucumbers, tomato, hummus, yogurt, and parsley, if using, with lemon wedges, sea salt, and more oil alongside so everyone can assemble their own flatbreads as desired.