There is a hierarchy in the parent/child relationship. As I often say to clients, adults have to be adults so that kids can be kids. They need to know that grown ups are looking out for them as they venture out into the world more and more. A parent is not their child's peer. There may be times when they can be a friend, but first and foremost, they are a parent. They are responsible for keeping their children safe and provided for until the child becomes a grown up. Kids are dependant on their parents for food, clothing, shelter, and emotional needs.
Parents have authority to make rules and set limits, to grant or revoke freedoms, allow certain behaviors, to take away or extend privledges, and so on. It is not an equal relationship. This doesn't mean parents should be bullies and power trip just because they can. The parents' role is to guide and set limits, to be firm but flexible, not to dictate or rule with an iron fist. I believe the role of the parent is to provide a balance of nurturing and discipline, to model responsible behaviors and communication/conflict resolution skills, and to equip kids with tools to manage feelings and make healthy decisions.
As kids get older, they are granted freedoms to make smaller choices so that, when they are teenagers and young adults, they are prepared to make choices about the more important things. By the time people reach the age of college, career, marriage, and having their own kids, if parents have done their jobs well, their offspring are usually ready to be "launched." In other words, we hope that, once they've gone through the innocense of childhood and the rebellion and limit-testing of adolescence, young adults will be adults in age and in maturity level. We want them to be able to take on grown up responsibilities and challenges. This means paying their own bills, having a job, buying groceries, and remembering to set up their own dentist appointments. True adults are able to make their own choices, have distinct ideas about what they want from life, career, and relationships, and don't expect someone else to be responsible for them.
So, when people who are legitimate adults both in age and maturity decide to enter into a dating or marriage relationship, they are going to seek out someone who is also functioning as a "for real" grown up. There is not a parent/child or adult/child dynamic of any kind. It is an adult to adult relationship. They enter the relationship as equals. There is not a hierarchy. They are peers and neither of them is in a "one up" position to have more power or authority or say in the relationship.
This means neither person feels they have the right to try and push personal agendas or control their partner. They are looking for someone who will be their partner, not their parent. They choose to get involved in a relationship because they want to be with the other person, not because they have to be. They aren't with their partner to make someone else happy, to have a roof over their head, or because they need to be a rescuer or to be rescued.
In an adult to adult relationship, both partners are willing to take responsibility for their own choices and behaviors. They can each decide for themselves what they will and won't do or what they will or won't tolerate as far as the other person's behaviors.