Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low, cover, and simmer until tender, about 20 minutes. Drain and cut into cubes.
Combine mayonnaise, Parmesan cheese, Dijon mustard, vinegar, cayenne pepper, salt, black pepper, and paprika in a large bowl. Add potatoes, red onion, and green onions; gently toss to combine.