There are multiple factors that determine our personalities, with our parents being one of them. We are constantly in a process of socialization, in which every social situation we encounter (whether we are conscious of it or not) has some kind of impact on our behavior and our self.
I think there are certain traits we inherit from our parents. For example, I saw a question the other day with a guy asking why his girlfriend was always buying new underwear. It could have been a number of reasons. My theory was that she did it because, psychologically, she watched her mother do that, too.