ETA: I think ordinary parents are too-optimistic identifiers, and highly gifted parents too-pessimistic if anything.
I completely agree with this..
Any speculation about why this is true, if it is?
In addition to what's already been said on this topic, I would add that everyone wants their child to be great, but everyone also wants their child to fit in. It's slightly illogical, when you say it that way, but I think you see it in all sorts of areas. Being smart is good, but being profoundly gifted is odd. Being great at baseball is acceptable, but being great at fluid reasoning is not. So, instead of bragging about it, (also not acceptable) we try to explain to ourselves ways that make it not so.
And, jumping to the original topic: I think the title is a bit misleading. The parents are agreeing to statements which in turn identify the child as gifted. They are not themselves identifying their child as "gifted" and snatching that word out of the air. Most of us only realized that our child(ren) were "different", and it took us a while to get to the reason. I also think that if you were just asking people if their child is "gifted" you'd get a higher than appropriate response. After all, "everyone's child is gifted." (sorry, I loathe the term gifted; I have a fondness for precision in language) If you asked if they thought their child was in the top 3% you'd get a lower than appropriate response, because 3% seems far above the acceptable normal range of smart.
Speaking to the normal-for-us idea -- that was totally us -- it sounds a bit bad to say it, but we expected DS to be smart, even really smart, because, well, he's our kid. But, wow. After a while even DH couldn't deny it, and he's the best at spinning things. On the flip side of that, I find myself pointing out to my friends that their kids are really smart, and even likely quite gifted, and that they shouldn't use DS as the benchmark of what gifted looks like because normal for our group isn't average.
