I don't think parents are excellent identifiers of giftedness. I often see numbers quoted to support the notion, but don't recall seeing any credible numbers about how many parents misidentify their kids as gifted. I think the internet has hugely intensified pushy-parent syndrome, so that if a child is not obviously delayed there's a bigger chance than ever that her parents will think she's possibly gifted.
I think this happens for lots of reasons. Talking about giftedness is trendier than ever, prompting many over-competitive types to focus their unholy energies in that direction instead of botoxing their children for the latest pageant, forcing their kiddos to run marathons, etc. Over-focusing on over-simplified milestone lists, especially with confusion over just how much early development truly correlates to giftedness, can only magnify this trend. At least in the US, concentration of needed resources in magnet schools, where there is typically some sort of bar to entry, adds incentive and angst. Etc. etc. etc.
ETA: I think ordinary parents are too-optimistic identifiers, and highly gifted parents too-pessimistic if anything.
