Is it possible to pass herpes from a cold sore from a child to the parent?

Yes. Herpes labialis can be spread from person to person. Parents have uasually already had an outbreak themselves some time in their life. It is a very common infection. If you already had it you can't catch it again (new) but you can have your own outbreak because it stays with you the rest of your life. So the child can spread it to you only if you have never had it before in your lifetime.