Re-experiencing childhood. Regardless of how good your own childhood was, your child allows you to vicariously relive childhood. Usually it's much better the second time around, when you have all the wisdom of age but can enjoy the energy of youth. Playing with a two-year-old feels like being two years old again.

I do not see it as re-experiencing childhood entirely, although there is a vicarious component of going to the zoo, christmas, etc..
For me it is experiencing the pure joy of childhood as understood by an adult. As adults we only have our memories of childhood, but in our children we celebrate their childhood with them. We can witness, take part and appreciate what they will only fully appreciate when they are much older.