Make dressing: Whisk ingredients together in a small bowl.
Marinate chicken: Put meat and 1/2 cup dressing in a medium bowl; let sit 30 minutes at room temperature. If using bamboo skewers, soak in water 30 minutes. Add enough water (2 to 4 tbsp.) to the rest of the dressing to thin slightly, then chill. Prepare a grill for medium heat (350° to 450°).
Assemble kebabs: Thread 1 peach wedge and 2 chicken chunks onto a skewer. Repeat, then add a final peach wedge. Thread remaining skewers the same way.
Cook meat: Oil cooking grate, using tongs and a wad of oiled paper towels. Grill kebabs, covered, until browned and chicken is cooked through, about 12 minutes, turning once with a spatula.
Make salad: Arrange arugula on plates and sprinkle with onion and parsley. Lay a kebab next to each salad and serve with remaining dressing.
Note: Nutritional analysis is per serving.