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