I recalled a time in middle school after I first moved to America. This kid I don't know was attempted to cuss (a supposedly monumental moment in every child's life, I'm sure) at me, but unfortunately I was not talking to him to begin with. So when he shouted 'Your Mother!' at me all of a sudden, I did not receive the insult as most would. Do not be mistaken, there IS a Chinese-equivalent for 'Your mother', and it most likely means the same thing in this context. However, as I was translating it in my mind, I was more distracted by the fact that he didn't say 'Your Father' at me. 'Your Father' is actually an existing insult in Chinese, and it holds the same meaning but only for the other member of your family.
Thinking back on it today, I started talking to my mom about it. We joked about how the reason why Americans don't seem to have much insults directed at the paternal member of the family is because if you dared to insult the man of the house, you'll most likely be chased down by a rifle. Then, mulling over some dish-washing, I realized how sexist that was. While we were pointing out and making fun of the way American insults are belittling the role of women, we were unconsciously placing the males in a higher rank in terms of power by justifying the reason 'Your father' is not a common saying. I guess next time someone attempts to insult my mother, I'll be unable to think of anything except this incident, and maybe snicker a little when I deem the situation safe enough to do so.
During our conversation, my mother told me that the reason such insults were so effective in the past was because if a family were to lose a parent, they would suffer greatly at the lost of financial support. Hence why the related insults such as 'Death to your mother/father' are so effective, and also why the male is included in the Chinese version of this insult. I did a brief wiki-search about 'Your mother', and apparently it focuses on the 'widespread sentiments of filial piety', of which becomes offensive to all women. In a shallow conclusion, I was able to guess that while they both serve similar purposes, the cultural differences between the Chinese 'Your mother', and it's American counterpart are vast.