Earlier quoted context omitted.
I imagine most would agree that the phrase "live through their kids" has a negative connotation and "caring about someone" does not.
Fair enough, I'm probably in minority. I don't see anything wrong with living through the kids, for me it's just one way to explore the world.
Where it starts being wrong is when people take "living through my kids" as "controlling and demanding my kids live the way I imagine I would have wanted to live, if I was in their shoes now".
Note: when I say "kids", I mean someone's children who are already adults and have their own life to live. I am not trying to say that kindergarten-aged children should be given full freedom to make their own life decisions and that any attempt to infringe on that is wrong.