I've never heard of the "parent function" before. (Other than changing diapers. ) So I Googled it and here is the list I got...
Of particular interest to me, was this site - that at least said, "There is no such thing as a "parent function." Recently, with increased use of graphing calculators and transformation geometry, math educators have called the most basic functions [those listed on the face of the calculator and some others] parent functions. Functions which include or are derived from these functions are called related functions."
"parent function"
Finally, another one of the links gave this answer to the question of what is the parent function for the exponential function...
WikiAnswers - What is the parent function for the exponential function
I honestly don't like this description ("parent function") as I find it confusing and it doesn't really add anything substantive to the field. Calculators differ widely in what functions are listed on their "face" and therefore this description is almost useless - if not downright harmful because of its ambiguity.