Put onions into a large bowl of cold water. Swish onions with your hands until water becomes cloudy. Transfer onions to a clean bowl of cold water, and let soak 15 minutes. Repeat twice. Drain, and pat dry. Transfer to a large bowl.
Whisk together vinegar, honey, lemon juice, mustard, salt, and pepper in a bowl. Add oil in a slow, steady stream, whisking until emulsified.
Add radicchio, lettuce, tomatoes, and parsley to bowl with onions. Drizzle with dressing to taste, and toss. Serve with remaining dressing on the side.