To make Port steak sauce:
Boil balsamic vinegar in a large, wide frying pan over high until reduced by half, 2 to 3 min. Add remaining ingredients to pan. Continue boiling, whisking often, until slightly thickened and reduced to 1 cup, 2 to 4 more min. Cool completely.
To make Port grilled mushrooms:
Preheat barbecue to medium-high. Grill whole mushrooms until cooked through, basting often with Port Steak Sauce. Transfer to a platter and sprinkle with chopped parsley.