Whisk 2 tbsp olive oil, lemon juice, garlic, oregano, salt, and pepper. Pour over the chicken, turn to coat, and marinate 15 minutes while the grill heats. (Even 15 minutes makes a real difference.)
Heat a grill or grill pan to medium-high. Toss the asparagus with the remaining 1 tbsp olive oil, salt, and pepper.
Grill the chicken 6-7 minutes per side until cooked through (165°F inside). Rest 5 minutes — the juices redistribute instead of flooding the board.
Grill the asparagus 4-5 minutes, turning once, until tender-crisp with charred spots.
Serve the chicken over the asparagus with lemon wedges and parsley.