Fill a large pot with salted water and bring to a boil. Add potatoes. Reduce heat to medium-low and simmer until fork-tender, 10 to 20 minutes. Drain immediately and rinse with cold water. Set aside.
Combine avocado, oil, parsley, onion, garlic, lemon juice, and mustard in a blender. Blend until dressing is smooth. Thin, if necessary, by adding 1 tablespoon of water at a time.
Transfer cooked potatoes to a large bowl and pour dressing on top. Gently mix until thoroughly coated. Season with salt and pepper.
