Cut crosshatch pattern on mushroom tops.
Mix olive oil, red wine vinegar, garlic powder, 1 tsp salt, and 1/2 tsp pepper in a zipper-lock bag.
Add mushrooms, seal bag, coat, and marinate 30 mins to 1 hour.
Combine feta, roasted red peppers, and sun-dried tomatoes.
Whisk mayonnaise and parsley separately.
Insert toothpick in each onion slice.
Heat grill until hot.
Clean and oil grate, brush onions with reserved marinade.
Grill mushrooms and onions, gill side up, 4-6 mins.
Flip and grill another 3-5 mins.
Remove onions, discard toothpicks.
Top mushrooms with feta mix, grill 3 mins until heated.
Grill rolls cut side down, 1 min.
Spread mayo on bottom buns, top with mushroom, onion, and arugula.
Add top buns.
Serve immediately.