That, is exactly what they're thinking. Embarrassing the child is exactly the purpose. After all, one has to start early building up blackmail material against their own offspring! How else will they successfully manage to embarrass their child when s.he brings home their partner in years to come?
Becoming a parent pretty much gives us the opportunity for payback. We all have that embarrassing photo where we're butt naked in the bath tub, or wearing some silly outfit, or perhaps even wearing a bowl - or Gods forbid, our own potty - as a hat! If you're lucky enough to have a sibling, they may take advantage of these photos and use them as blackmail material against you themselves! Maybe even end up putting them online! The horror!!
This, is why parents do it. It is a vicious cycle of sorts.
Hell, I know I look forward to it myself!