Spread yogurt in a circle on a large serving plate.
Combine parsley, vinegar, oil, honey, lime juice, 1/8 teaspoon salt, black pepper, and red pepper in a medium bowl, stirring with a whisk. Add all melon balls and radish to bowl; toss gently.
Using a slotted spoon, mound melon mixture on top of yogurt (reserve vinegar mixture). Sprinkle evenly with bacon and remaining 1/4 teaspoon salt. Drizzle with 2 tablespoons reserved vinegar mixture; discard remaining vinegar mixture. Sprinkle evenly with basil leaves.