I got ahold of the original research article. Here is their test of overvaluation (responses are on a scale of 0-3):

Quote
1. Without my child, his/her class would be much less fun.
2. My child deserves special treatment.
3. I would not be surprised to learn that my child has extraordinary talents and abilities.
4. I would find it disappointing if my child was just a “regular” child.
5. My child is more special than other children.
6. My child deserves something extra in life.
7. My child is a great example for other children to follow.
They compared the results against personality measures of the parents, and, crucially, an objective measure of the parents "overclaiming" about their kid's knowledge. (This last was a pretty clever test. They asked the parents to rate their child's familiarity with various items from history, geography, and literature. Included on the list were some fake items. The extent to which the parent rates their kid as highly familiar with the fake items is the measure of "overclaiming.")

It's pretty clear that the research is not about parents who accurately evaluate their kids' unusual abilities, but rather, parents who want to believe that their kid is unusual, and are caught up in a narcissistic viewpoint of being "special" and more "deserving."

(Personally, the only item I would score high on is #3. I'll cop to a slight needle-twitch on #4, but that's because I love it that my child is weird like me. Still, that only gives me a score of 4 out of a possible 21.)

As usual, the popular press has missed the point. This isn't about, "bad parents, you're all over-evaluating your kids!" There's really no news here -- we all know that there are a few parents who are "like that." This research article is actually a fairly dry technical piece introducing a new measure for identifying parents who are "like that," and validating it (showing that it actually tracks what it is supposed to be measuring).