Heat gas or charcoal grill. In 10-inch skillet, heat oil over medium-high heat. Cook garlic in oil about 30 seconds, stirring constantly, until fragrant. Add bread crumbs. Cook 2 to 3 minutes, stirring frequently, until toasted. Remove crumb mixture to small bowl. Stir pesto and Parmesan cheese into bowl with crumbs.
Loosen skin on chicken to form a pocket, without detaching skin completely. Fill each pocket with pesto mixture. Sprinkle with salt and pepper.
Place chicken on grill, skin side down, over medium-low heat. Cover grill; cook 10 minutes, moving occasionally as needed to avoid burning the skin. Turn chicken; spoon any remaining pesto over chicken. Cook 10 to 15 minutes longer or until juice of chicken is clear when thickest part is cut to bone (170°F).