In a large bowl, combine the minced garlic, lemon zest, lemon juice, olive oil, salt, and black pepper. Add the chicken breasts and turn them to coat evenly. Let the chicken sit in the marinade for 15 minutes while you prepare the grill.
Preheat an outdoor grill to medium-high heat. Place the chicken breasts on the grill and cook for 6 to 7 minutes per side until the internal temperature reaches 165°F. Remove the chicken and let it rest on a plate.
While the chicken rests, add the trimmed asparagus to the same bowl and toss it in the remaining marinade. Place the asparagus directly on the grill grates and cook for 4 to 5 minutes, turning once, until it is tender with light char marks.
Slice the rested chicken and arrange it on a serving platter with the grilled asparagus. Sprinkle everything with the chopped fresh parsley before serving.