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It wouldn't be the parents making the errors, it would be the clerks at the Social Security administration who put the data into the computer. Entering the sex code into the computer probably originally involved typing in either a "1" or a "2", and it would have been very easy for a mistake to have been made. I have no idea why there would be more errors in the 1980s than in the 1970s or 1990s, but the accuracy in something like this might depend partly on how well trained the individuals involved were. It's quite possible that the 1980s data were actually entered first, with the 1970s data only being entered in later along with the 1990s while the system was becoming completely computerized.

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I certainly haven't had any direct personal experience of a boy named Lizzie; but then I hardly know every person in the USA! :)