In a small bowl or jar, whisk together the olive oil, red wine vinegar, lemon juice, minced garlic, dried oregano, salt, and pepper until well combined. Set aside.
In a large mixing bowl, add the rinsed and drained cannellini beans, kidney beans, chickpeas, diced cucumber, quartered artichoke hearts, and chopped sun-dried tomatoes. Toss gently to combine.
Pour the prepared dressing over the bean mixture and toss until everything is evenly coated. Fold in the shaved Parmesan cheese and chopped parsley.
Cover the bowl and refrigerate for at least 15 minutes to allow the flavors to meld. Serve chilled or at room temperature, garnished with extra Parmesan and parsley.