In a large pot, cover potatoes in cold water. Bring to a boil over medium high heat, and cook for 5 to 8 minutes, until tender. Drain, place in a large bowl, cover and refrigerate until cold.
Mix together the mayonnaise, olive oil, garlic, pesto, spinach, vinegar and salt with a whisk.
To the bowl of chilled potatoes, add red peppers, tomatoes, basil and parmesan. Pour dressing over and toss. Serve cold, or refrigerate until ready to eat.