Mix first 5 ingredients and 1/2°C parsley in a large bowl to blend. Transfer crab mixture to 8x8x2-inch glass baking dish. You can prepare up to this point and cover and refrigerate for 6 hours.
Preheat oven to 400°. Bake crab mixture until bubbling and heated through, about 25 minutes.
Preheat broiler. Place bread slices in a single layer on a large baking sheet. Divide hot crab mixture among bread slices, using about 1/2 cup each. Top with tomato slices, then Monterey Jack cheese slices. Broil sandwiches until cheese melts, watching carefully to avoid burning, about 2 minutes. Sprinkle with the 2 T. of parsley and serve hot.