Preheat your grill to medium-high heat.
In a small bowl, whisk together olive oil, lemon juice, oregano, garlic powder, onion powder, salt, and pepper.
Place the chicken breasts in a large resealable plastic bag or shallow dish and pour half of the marinade over them. Seal the bag or cover the dish and marinate in the refrigerator for at least 30 minutes, or up to 2 hours for more flavor.
While the chicken is marinating, toss the sliced zucchini and squash with the remaining marinade in a separate bowl.
Remove the chicken from the marinade and discard the used marinade. Grill the chicken for about 6-7 minutes on each side, or until the internal temperature reaches 165°F.
During the last 5 minutes of grilling, add the zucchini and squash to the grill. Cook until they are tender and have grill marks, about 3-4 minutes per side.
Once cooked, remove the chicken and vegetables from the grill. Let the chicken rest for a few minutes before slicing.
Serve the grilled chicken with the zucchini and squash, garnished with fresh parsley.