To understand this expression, it helps to know that the sport of curling is much more popular in Sweden than in the U.S. You don’t need to be an expert on the sport to get the joke, though.
If you’ve ever seen a clip of a match from the Olympics, you can probably imagine the way the curlers scrub at the ice with their brooms, clearing a path for the stone. Curling parents are just as intent as they try to smooth the way for their precious offspring.
Ever since I heard about curling parents, I’ve lamented the fact that there was no equivalent expression in English. “Overprotective” isn’t quite the same thing, and “over-involved” isn’t nearly as vivid.