Hi Tony,
There are pros and cons to every decision, but we've made the decisions that living people must be private, and the names of private people's children are not revealed. (If your father were non-living, as you seem to know, your parents names would be revealed if you were "Private with a Public Family Tree" but since his profile is also private, and you are his child, your connection is hidden to everyone except on those on Trusted List.)
Chris