If you did more that 1 second of research, you'd know the history of the middle class in the US starts in the early 1800s with the growth of manufacturing, not WWII. WWII was a huge boost for various reasons (women in the workforce, GI Bill), but it doesn't negate the parents point about the immigrants of the previous century.
We did also get some hundreds of thousands of immigrants (boat people) in the 1980s, I think mostly without suitcases. My impression (casual) is they have done fairly well.
The persistence of this myth is truly remarkable. Consider that Veblen published "The Theory of the Leisure Class" in 1899. What we consider today the middle class was very much alive and booming by the end of the 19th century.
Hint: start with WW2