mommamia22- I’ve heard that concept of not apologzing directly, but never with a reputable source. Until then, I strongly disagree. Children are no less human than anyone else.
If you did something wrong you need to ask mechila like everyone else. From a chinuch standpoint its great to discuss how you will react differently next time so your child can learn to do the same.
But dont you see how much more you can teach them by apologizing? Some times kids feel that parents give them no respect -another thing that some parents wrongly feel they are allowed to do- and dont care about their feelings. Asking mechila will boost their self respect- and respect you more for it. They will see how hard it was for you to apologize and learn to do the right thing even when it is difficult. (I feel very strongly about this.)