The "American dream", whether or not you believe it to be true, is essentially that you can come to America (ie the U.S.) with very little, work hard, and make a better life for yourself and/or your children due to the opportunities available here.
I would interpret "Russian dream" from the context to be both a parody of the American expression and, since it's an item of clothing, to also mean that the girl wearing it is a dream; in other words, that she's very pretty.
Yljaha, it really is a very funny parody and I think Mike, and your husband, have got the joke just right. I've been laughing to myself all morning since I read it.
Actually parody is one of the high points of humour and I think it is a great thing to wear.
"a pullover with THE Russian national pattern"
"parody on THE expression"