How can I be both a parent and a friend without compromising my authority?

There is no way to be a friend while you are parenting. You are not their friend. You are their parent only, and that is the way it should be. Friends do not have authority. Only parents do.

It helps your relationship in that way if you are empathetic and understanding of the emotions and thoughts they are having, and make time to spend with them doing things they enjoy. However, you are the parent. Other people can be their friend, but no one else should be their parent. Because being the parent is your job and being their buddy isn't, sometimes you just have to let go of trying to feel like their friend. They may not enjoy you doing your job as parent right now, but that will likely change as they mature.

So I work with children as my job and passion-project, and I've dealt with this extensively. I've always been really frustrated by the mantra of "your children are not your friends", which is often used as an excuse to mistreat children. A mentor once gave me a nice addendum to that statement, which was, yes, you're right, they're not your friends, since you're not social peers and as an adult, you do have power that a child doesn't have, but that doesn't mean you can't *be friendly*. There are kind ways of being authoritative. Those two things don't have to be mutually exclusive.