Prepare the grill for indirect cooking over medium heat (350° to 450°F).
With a large knife, carefully cut each squash in half lengthwise. Scoop out and discard the seeds and strings.
In a small saucepan over medium heat, combine the butter, brown sugar, sage, granulated onion, maple syrup, salt, and pepper and cook until melted. Brush the butter mixture all over the exposed flesh of each squash.
Brush the cooking grates clean. Grill the squashes, cut sides up, over indirect medium heat, with the lid closed, for 45 minutes. Sprinkle the cheese and parsley all over the exposed flesh of each squash. Continue to grill until the flesh is very soft when pierced with the tip of a knife and the cheese has browned, 20 to 30 minutes more. Serve warm.