Preheat the grill to medium heat. Pat the chicken pieces dry with a paper towel, then oil, and sprinkle with salt and pepper.
Mix the 2 tablespoons oil, both mustards, maple syrup, parsley, 1/2 teaspoon salt, and 1/4 teaspoon pepper together in a small bowl.
Grill the chicken over indirect-medium heat for 15-20 minutes, then flip and brush with mustard glaze. Grill another 15-20 minutes and repeat—40 minutes total.
If roasting in the oven, preheat to 375 degrees F. Brush with glaze and roast for 45 minutes.