A parent function is the original function (before the transformation).
There are probably better/correct answers than this...
I agree with Gh0sty15 that it is hard to explain if you do not know what it is, but I will give it my best shot!
A parent function is the simplest representation of a family of functions.
For example, let's figure out the parent function of a linear equation. It is:
As you can see above, this is the simplest way to represent a linear function. Now, let's try to represent a quadratic function.
The above is the least amount required to get a parabola when it is graphed.