If we're being truthfully honest, no parent really wants to apologise to their kid thinking that it will damage their authority 

• It models good behaviour- No parent can expect their child to apologise when they do something wrong if their parents will never do the same. You (as the parent) are the role model so be a good one and behave how you would like your kids to. Teach them to react to situations in the way you believe everyone should and accept that a big part of that, is saying a simple “sorry” when you have done something wrong.
• You made a mistake and feel you need to own up- Parents are not showing weakness by apologising to their kids as everyone makes mistakes and your kids will relate to you more. If apologising comes easy it will cause less conflict at home as one of the biggest issues in any home is people refusing to apologise and therefore a family dispute lasts much longer.
• Shows mutual respect- A child will gratefully accept a heartfelt apology and will consider it respectful to apologise. This means you and your child will have full respect for one other and in turn for other people too. Your child will continue these good manners when they are out and about away from you which means you can trust that they will be good and apologise if they need to.
• It makes you feel better- Apologising to your kid makes you feel better because you may have made an error of judgement and it may have had some consequences. Parenting isn’t easy and it is on the job training and learning, but at least when you do mess up you can put your hands in the air and admit you were wrong while apologising.
• You lost it and said things you shouldn’t have- A child who has been given out to for no apparent reason will feel hurt and sad. The fact of the matter is parents have bad days, work may be stressful and perhaps you have a sick relative. It is easy to snap at those closest to you and your little people are about as close as you can get! Losing your temper is natural at times but knowing when to be the bigger person and apologise to your kid is vital. Telling your kid that you were stressed and admitting you shouldn’t have yelled at them will keep you both bonded.
• Everyone needs to apologise sometimes- Your child will go to school, have teachers and eventually have jobs and do you think they will get through it without ever admitting they made a mistake or by never having to apologise? Teaching your child now to apologise will set them up on the right path for the years ahead.