In the Modern Astrology of the twenty century, the Sun represents the father archetype (not the biological father, because a woman can also express solar characteristics), one who has free thinking and independent decisions taking, refusing external interferences.
Saturn represents the mother archetype, one who cares for security and fisical protection.
And the Moon represents the child archetype, one who wants to have fun and has spontaneous pleasant and unpleasant feeling to environmental experiences.
I think the Sun has to do with the father in my case. It may not in your case. It definitely does in my mom's and her siblings,and other people's charts that I have looked at. I guess we will have to agree to disagree. I am very sure that Moon has to do with the mother.
I believe the Sun has to do with the ego,self expression,and vitality as well as the archetypal male.