Why do we (some of us, anyway) actually try to embarrass our teens?

In a question about ‘cool’, @Darwin writes, ”I definitely get enjoyment out of embarrassing [my teens], especially in the grocery store.”
“Oh, Mom!”

I have to admit that I enjoy embarrassing my kids, too, by highlighting just how uncool I am. I think I’m making fun of the cliche that parents embarrass their teens, but I’m also trying to get my kids not to take their coolness so seriously. Unlike me, my kids actually are cool, or at least, they think they are. In any case, they enjoy a popularity that escaped me at that age.

So what’s going on here? Is this an intergenerational coolness fight? Are we all trying to advocate for our style or way of being in the world? Is it all tongue-in-cheek (at least on the adult side)? Or is there some underlying reason (maybe even jealousy) that drives us to be embarrassed by each other?