Place chicken breasts between 2 sheets of heavy plastic on a solid, level surface. Firmly pound with the smooth side of a meat mallet to a 1/4-inch thickness. Remove top sheet of plastic. Cut each chicken breast into strips. Roll up and thread on a skewer.
Combine parsley, Dijon mustard, Parmesan cheese, basil, and garlic in a small bowl. Spread mixture over skewered chicken.
Place chicken skewers on a baking sheet and cover with plastic wrap. Refrigerate until flavors combine, at least 2 hours.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Uncover skewers and broil in the preheated oven, turning once halfway through, until chicken is no longer pink and tender, 10 to 15 minutes.