When I was younger, my parents used to have all these stories ranging from forgetting to take birth control to overdue tax bills. When I was not busy laughing uncontrollably or gasping in horror, I think I learned a lot from what they told me of how they dealt with it. Of course, these only started when I hit thirteen years of age. Personally, I think the age recommended to tell kids they are adopted (2-4) is the time to start introducing some fail stories.
There is a problem with the Golden Child era of parenting. There are multiple children who reveal themselves to be less than stellar once they leave their mother's comforting arms, and when these children start to stagger and struggle, it is the natural reaction of the parent to blame others. This is selfish and wrong and only leads to children not learning from errors.
I think that it is helpful for children to understand their parents mistakes. It is helpful to a child to learn through their own mistakes, but also learn from their parents. Knowing that a parent is not perfect and has made mistakes can take pressure off a child to perform perfectly. Without the pressure of living up to a perfect parent who has made no mistakes, a child can breathe a little easier. It also gives a chance for a parent and child to relate to each other on another level. Seeing your parent as a person and not just a parent can bring them closer together.
Children can definitely benefit from their parents mistakes, but only if parents use them as teaching tools. However, there is a time and a place for it. The child need to be intellectually and emotionally mature to be able to listen and understand. So, most teenagers would not be ready to hear such stories.
Parents do not need to tell their child every little mistake they made in the past. The parents should just express their concerns to the child and tell them to do what is right. If the child knew all the bad things the parent did, it could hurt the kid.
I don't think they should. A parents job is to guide and teach their children based on their own experiences of life. This includes being influenced by their own mistakes. However I don't think a parent needs to actually explain all their previous failings. This would not benefit the child, and only undermine the parents authority.