Preheat grill to 400˚F / 205˚C. Cut squash in half lengthwise, then use a spoon scoop out the seeds.
Brush each half with ½ tablespoon of olive oil.
Sprinkle each half with ¼ teaspoon of salt and some freshly ground black pepper.
Grill cut side up on indirect heat for 40 minutes, then grill cut side down for 5-10 minutes.
Remove from the grill, sprinkle with the fresh parsley (if using), and let cool for 5-10 minutes.
Use a fork to hold the squash in place while you use another fork to shred the squash. Place in a bowl with the remaining tablespoon of olive oil, and a touch of salt and pepper and stir to coat.