Bring 2 pots of salted water to a boil. Cook the white potatoes and purple potatoes separately until tender but still firm, about 15 minutes. Drain and cool. Transfer to a large bowl.
Meanwhile, mix the vinegar, garlic, salt and pepper in a mixing bowl. Drizzle in the olive oil, whisking until mixture emulsifies.
Toss the potatoes with the vinegar and oil mixture and then toss in the arugula. Toss gently so the potatoes do not break up. Sprinkle with salt and shave the Parmesan over top of salad. Serve at room temperature. Enjoy!