On Thursday morning, I inexplicably, irrationally, and ultimately, cruelly, lost my temper at my four-year old daughter; I wanted her to do X; she did not; I thought my request was reasonable; she didn’t think it was; and then, when on my demanding reasons for her decision and denial of my request, she could not comply, I snapped. I stormed off, fuming; she was left in tears. Even as I did so, I knew I had fucked up, and spectacularly. And yet, perversely, my irritation and frustration–which was really what my anger amounted to–continued to cloud my mind for a minute or two. As those feelings receded, I walked back into my daughter’s bedroom, picked her up, gave her a hug, and asked her if she was hungry and wanted breakfast. She perked up, and said she did. A second or so later, as I carried her into the kitchen, she said she was ‘sorry’; I said I was too; and we hugged again. A minute or so later, she was smiling and happy. (Her mood improved even more when I told her I would get her a ‘pizza treat’ later that evening.) An hour later, she had left for preschool, and I headed to midtown Manhattan to get some work done at the CUNY Graduate Center library.
But all was not well; I was beset with a series of nagging thoughts all day. My daughter hadn’t done anything wrong; she had said ‘sorry’ because she knew a parent was angry at her, and that’s what you do when your parental figure is upset with you. I had been in the wrong all along; once my initial request had been denied, I should have backed off. Instead–like a petulant child–I had insisted, and then later, browbeaten her with a series of badgering demands for clarification of her reasons, all the while intimidating her with my tone of voice and body language. My daughter had never needed to apologize; she should have demanded one from me. I was the offender here; my perfunctory apology and ‘make-up’ in the morning was not enough.
That evening, I picked her up from pre-school, bought some pizza, and we returned home to eat and watch–as promised–a couple of short videos on lions and tigers in the wild. As we ate, I offered a more elaborate apology: I said I should have listened to her and respected her wishes, that she had been right, and I had been wrong. She listened rather solemnly–or about as solemnly as four-year olds can–and on my asking if she understood what I was trying to say, nodded her head. We then went back to watching big cats do what they do best.
I knew there would be times when I would have to apologize to my child; error-free parenting is impossible. I’ve done so before, but I don’t think I’ve ever quite made my admission of wrong-doing quite as explicit as it was on this occasion. Truth be told, it was a curiously uplifting experience.
Samir Chopra 
I do not know how time constrained your job is. Very rarely my wife and I had to be at work at a specified time. We could wait until our son complied or the issue became moot. When our son was in kindergarten, one day he refused to eat Idly for breakfast, but like all others in his class wanted sugar loops. My wife and our son had a silent tug of war that ended my son agreeing to eat Idly only if it was coated with brown sugar. My wife gave him the impression that she gave in to his wish. This was possible because she had time to negotiate.
I wrote a short story on that incident.
Not to say that you did wrong by apologizing to your daughter, I will say, that having repeated your little interaction more times than I care to admit, when I look back I realize that from the perspective of a 4 year old, the apology is seen as more parental talk, and she doesn’t get what you are trying to convey. However, it is important to be sensitive to our kids feelings and not to take out our shit on them. Parenting is a humbling experience.