For the marinade: Add the parsley, vinegar, oregano, red pepper flakes, scallions and garlic to a food processor and pulse a few times to combine. With the food processor running, stream in the oil until uniform in a consistency. Add the lemon zest, 1/2 teaspoon salt and several grinds of pepper. Pulse just to combine. Add 4 boneless skinless chicken breasts or 8 boneless skinless thighs (about 1 1/2 pounds) to a large resealable plastic bag. Pour in the marinade, squeeze out the air, seal tightly and massage gently to coat the chicken.
Refrigerate for 2 hours or freeze up to 1 month; defrost in the refrigerator overnight before cooking. Save some marinade for serving with the cooked dish.
To cook: Grill or transfer the chicken to a foil-lined baking sheet and roast in a 425-degree F oven until cooked through and an instant-read thermometer registers 165 degrees F, 20 to 25 minutes. Serve with reserved Chimichurri marinade.