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 2 Tbsp., 18 to 20 minutes.
3. 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.
4. Grill, skin-sides up, covered, 15 minutes.
5. Uncover, baste with some of the glaze and grill, turning and basting occasionally, until cooked through, 10 to 12 minutes.
6. Sprinkle with fresh flat-leaf parsley and serve with remaining glaze.

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