Whisk olive oil and red wine vinegar in large bowl. Season dressing with salt and pepper. Mix tomatoes, cucumber, bell pepper, red onion, radishes, and parsley into dressing. Stir in feta cheese. DO AHEAD: Salad can be made 2 days ahead. Cover and chill.
Using slotted spoon, transfer salad mixture to pita bread halves. Serve sandwiches immediately.