Fire up your grill for indirect cooking 350-400 F.
Prepare your onions by cutting 1/2 inch off the stem end, then peel all the skin off. Place the onion on a cutting board cut side down.
Make cuts vertically thru the onion starting 1/2" from the root part that is sticking up. You'll make 16 evenly-spaced cuts around the onion, all the way to the cutting board. Make sure you go all the way through.
Carefully transfer the onion to your plank and turn over so the root is down. Carefully separate the pedals as if you were opening a flower.
Drizzle the onions with olive oil.
Sprinkle the onions liberally with the Italian seasoning, salt and pepper and Parmesan.
Place plank on grill away from heat and cook for 1 hour or until pedals have started to soften. If you prefer move plank over direct heat last 10-15 minutes to get more smoky flavor from the plank. Watch out for fires, though, and be prepared to move the plank back to indirect heat to put them out.
Garnish with parsley and serve.