Quester and I, and quite a few of our friends, practice a type of parenting that is different than that used by many in the generations just prior to us. Call it what you will. Attachment parenting covers it to some extent, though the basic gist of it is respecting your children as you would another adult human being (well, you have to have compassion to begin with, I guess—some people aren't very nice to other adults, either). My Mom was actually ahead of her time in this area. She has always said, “kids are just smaller people,” and they should be treated as such.
This doesn't mean that you aren't playing the role of the parent. When I was a teenager, I babysat for some neighbors who never told their kids “no,” and the two girls ran wild, making life very difficult for the parents. Kids need boundaries, and you are disrespecting them if you don't provide any, just as much as you are if you're arbitrarily strict.
The idea, in my view, is to create a healthy working relationship with your kids. Have fun with them, and also make it clear what rules you want them to live by. Teach them, and learn from them. Correct them when they mess up, then forgive them and move on; gracefully admit it when you're the one in the wrong. Because of your age and experience, you get to make decisions about their lives. But encourage them to make as many choices on their own behalf as possible, so they can learn to choose well.
Something I've learned from colleagues with adult children is that parenting never stops. In this culture, we either look forward to the day when the kids turn 18 and move out, or we lament the “empty nest” when they head off to college or work. But they are still your kids. You'll still worry about them, call them to see how they're doing, probably even loan them money. They'll distance themselves from you to some degree, as part of the natural process of finding out how to be on their own. But they'll draw closer once again, and you'll still be part of each others' lives. At least, ideally.
What I see from the parents of some of my friends, though, is a lack of respect that was probably there from the beginning. They treat their adult children as if they were still kids, meant to be “seen and not heard,” or told what to do. Rather than respecting the wisdom their son or daughter has gained, they instead belittle them, which is probably an old pattern, or take them for granted. This, understandably, makes the kids less likely to want to hang out with them. Some parents even refuse to be part of their adult kids' lives, due to some (imagined or real) slight or insult. What a waste.
What I'm learning from those relationships is how I don't want to treat my kids as they get older. In fact, perhaps the logic behind attachment parenting extends to the lifelong relationship I'll have with my kids. I hope that, together, we're learning how to be a family that can enjoy each other's company, work through our differences of opinion, and continue to love and learn together throughout our lifetimes.