Pound chicken breasts to about 3/4 inch thick and pat dry. Do not rinse raw chicken.
Stir Greek yogurt, 1 tablespoon olive oil, lemon juice, zest, garlic, salt, oregano, paprika, cumin and pepper until uniform.
Coat chicken, cover and refrigerate at least 2 hours and preferably no longer than 8 hours; do not exceed 24 hours.
Heat a clean grill to medium-high, about 400 to 450°F, with a cooler zone available. Clean and oil the grates safely.
Lift chicken from the marinade and scrape away thick ridges, leaving a thin coating. Discard all raw-contact marinade and brush chicken lightly with remaining oil.
Grill with the lid closed for 5 to 7 minutes without moving, until the chicken releases with gentle pressure.
Flip once and cook another 4 to 7 minutes, moving to cooler heat if needed, until the thickest part reaches 165°F.
Transfer to a clean plate, rest 5 minutes and garnish with fresh parsley and lemon if desired.