1: Heat grill to medium.
2: Combine blackberries, water, white wine vinegar, and sugar in a small saucepan; simmer, mashing occasionally, until liquid is reduced to about half, 18 to 20 minutes.
3: Stir in butter and whole-grain mustard.
4: Transfer half the glaze to a bowl; reserve.
5: Pat chicken thighs and drumsticks dry; season with kosher salt and black pepper.
6: Grill chicken, skin-sides up, covered, for 15 minutes.
7: Uncover, baste with some of the glaze and grill, turning and basting occasionally, until cooked through, 10 to 12 minutes.
8: Sprinkle with fresh flat-leaf parsley and serve with remaining glaze.
