My own mum pulled this one on me. We were talking on the phone and she could hear my daughter playing in the background. As one point it dawned on her that she could not hear my wife.

  "Is Susan out?"
  "Yeah she's gone out to [I forget what]."
  "So you're home babysitting?"
  "No, it's just Alice here, no one else."
  "???"

In an extremely polite but pointed way, I informed her that I was her father, not her babysitter. She never made that mistake again.

a lot of people? quite often it feels like everyone.

it's a stereotype reinforced by television shows, movies, commercials, etc. and it seems to sell (i'm assuming so, i figure approval boards to get these things made and shown wouldn't approve them unless it made them money).

it's a very sad statement about our society.

my father always says "if they're your kids, it isn't babysitting"

Saying to a dad that he is babysitting is so demeaning. It's like saying that his investment in his own child is on the level of that teenage girl next door that babysits his child every other week because she needs the 10 bucks to buy gas.