I would argue that one of the principal ways in which cultural conflict is expressed through this story is through the desire of the girls to assimilate and the desire of their parents to not allow them to "become Americans," which seems to be their goal. It is obvious from this story that the daughters are quickest to adjust to their new lives in America. They speak English better and more fluently than either of their parents, for their mother speaks an English that is a "mishmash of mixed-up idioms" and their father's English is worse than their mother's. Note the way that they normally approach their mother when they want to do something "American":
My sisters and I would seek her out now when she seemed to have amoment to talk to us: We were having trouble at school or we wanted her to persuade my father to give us permission to go into the city or to a shopping mall or a movie--in broad daylight! My mother would wave us out of her room. "The problem with you girls..." I can tell you right now what the problem always boiled down to: We wanted to become Americans and my father--and my motehr, at first--would have none of it.
Note the way that the conflict between the generations is presented. Even through something as simple as wanting to go to a mall, the desire to assimiliate and the desire to retain a distinct cultural identity are shown to be in conflict.