Hmmm...I had to read your post over a few time to get what you were really asking.
I think your definition of a "parent" is one of yelling and threatening and totally authoritative. In that case I can see why you are seeing a distinction between that type of parent and the type of parent/friend you are striving to be. I don't think you are your children's friend by your description. You mention being stern, correcting behaviors and keeping them home if they are misbehaving. A friend would not do those things...a parent does.
You are not parenting the way most of us grew up. I got the belt when I misbehaved, soap in the mouth when I disrespected or said a bad word, and I did not look to my parents for guidance or advice or to do something fun with. I turned to my friends...which usually led to more bad behavior. Friends commiserated about how awful and unfair their parents were.
My husband was raised the same way. We have determined to do things differently and it is working out soooo much better. We have read a couple books and attended parenting classes. We are seeing the positive results from this type of parenting instead of the " I will take you out to the woodshed if you cross me" type of parenting.
Don't take it personally if you hear that you are being your kids' friend when you don't yell or threaten. It is a childish behavior to yell and threaten...that is how kids deal with frustration and being out of control..not how mature adults act. You know you are correcting bad behavior and setting a good example of how to treat others and act in public. Parents tend to think that if the behavior stops in the moment because of a spanking or yelling that the child has learned the lesson and parent wins. Soooo wrong. The kid actually learns nothing but to avoid that behavior for fear of Mom or Dad's wrath. Sooo much more learning and behavior modification over the long run comes from a calm parent who teaches through more positive discipline.
I think you are doing fine. You are spending alot of time with your kids and in the future they will look back and be sooooo grateful for how you parented. I look back and wish my parents would have yelled less, spanked less, threatened less and instead talked to me, spent more time playing and reading and exploring with me. My siblings and I do not have close relationships with our parents to this day. I did not go to parents for advice about much of anything. There was a big disconnect and still is...but it is getting better now that we are equals.
If you see that your kids show that they can be kind, polite, work hard, can deal with others when things don't go their way then you are doing a good job at parenting.