In some parents' mindsets their children are almost a physical extension of themselves. For example, some, out of an unconscious desire for immortality name their kids by their own names.
But much more often there are those who simply identify their kids with that inner child in themselves. Then they almost literally "live" their kids' lives.
Depending on that parental (authoritative) figure in themselves (which they had likely inherited from their own parents) they are either over-protective or over-demanding parents to their kids.
For example, a perfectionist parent is bound to be over-demanding towards kids just as much as s(he) is over-demanding towards him/herself. We have all seen parents pushing their kids into competitive games, musical, or academic performance - living through them their own need for victories that they missed in their lives, pushing their kids to fulfill their ambitions.
Those others, who are extremely friendly and permissive towards their inner child are bound to be overly friendly and protective towards their kids. They are literally hovering like hawks over their offspring constantly making sure they are all right. More often than not it turns into an obsessive-compulsive behavior, as they find it impossible to resist those protective urges, rationalizing it with an explanation that they are just being "good parents".