Bring a large pot of salted water to a boil and cook the gluten-free pasta until al dente according to package instructions.
Drain the pasta in a colander and rinse immediately with cold water to remove excess starch and stop the cooking process.
In a small mixing bowl or glass jar, whisk together the olive oil, red wine vinegar, lemon juice, minced garlic, dried oregano, sea salt, and black pepper until well emulsified.
In a large salad bowl, combine the cooled pasta, drained chickpeas, diced cucumber, halved cherry tomatoes, sliced olives, and red onion.
Drizzle the dressing over the pasta mixture and toss thoroughly to ensure all ingredients are evenly coated.
Gently fold in the crumbled feta cheese and chopped parsley, taking care not to break the cheese down too much.
Cover and refrigerate for at least 30 minutes before serving to allow the flavors to fully meld.