Process vinegar, lemon juice, and garlic cloves in a food processor until smooth. With processor running, pour oil through food chute in a slow, steady stream, processing until blended. Add basil leaves and parsley leaves, and process until smooth. Add Parmesan cheese; pulse to combine. Stir in kosher salt and freshly ground black pepper to taste. Serve immediately, or refrigerate in an airtight container up to 5 days.