Yes perhaps it is the term "best Friend" that is the problem. I simply don't have a better one. I do discourage my young children from using the word because it can be hurtful to friends to call one of them "best". That should be reserved for your husband or wife.
All friends and family have special places, and there are different ones that you will want to share different things with. My point is that most parents don't have much of any relationship with their teenagers other than one of conflict. I believe that the parent-child relationship is special and sacred. It is many faceted, and different with each child.
My grown daughter and I have a special friendship because she is also a mother and we share so many beliefs and experiences. But of course her "best friend" is her husband....as it should be. As a teenager her peers were her closest confidants, yet there were and still are many things that she would rather talk to me about, because we understood each other better. Things like poetry, music, art, having babies and being a good listener to her friends. Sharing those things is what friendship is about, no matter the age.
My married son and I had a different friendship when he was a teenager. Now I consider his wife to be my friend. It would be "yucky" for my son to "confide" in me what should be reserved for his wife. Yet I would still consider them to be among my "best friends". He and I never had the same kind of relationship that my daughter and I did, yet it was still and friendship. His soccer buddies were his "friends" yet when he wanted to discuss deep spiritual matters he would seek out his father or I.
My 5 year old daughter is not the one I would look to when I need to share my worries over growing older, yet whe I want to play makebelieve she the first friend I will turn to!
Perhaps my definition of friendship is too broad. I simply do not have a better word. I will always consider my husband, my sister, and my children to be my closest friends. Through out life I have had many people that I have called friends. Most have come and gone. All have touched me, some very deeply. A few have held a lasting place in my heart. Yet, my family will always be there...they are the ones I will turn to in crisis. The ones I will share my sorrows and my joys with. To me that is frienship.