The reason is simple. She is just much more likely to be comfortable and more likely to see you as confident rather than inconsiderate. Just say "Hey excuse me, sorry to interrupt but but your friend here (pointing) caught my eye and I wanted to introduce myself."

Also, if she is with her friend then she is probably just hanging out rather than trying to get an errand done and go home. So she has time to talk. If they are in the middle of a lively conversation then don't interrupt but if not then you might be saving them from awkward silence.

With her mother is also fine (according to my mom). But not with her father (according to my dad). Fathers tend to be biased against guys that pursue their daughters. With their children is fine if you can see they don't have a wedding ring.