Unless the membership form actually has the words "from your Birth certificate" (I haven't checked recently) on it; then it would seem to me, for better or worse to default to at a minimum the legally recognized gender.
I do not know the details of NJ anti-discrimination law, but based on that they apparently allow a person to indicate their gender of affiliation on the legal document of a driver's license; that would imply that at least there, the parents would not by lying, but in compliance with NJ law when completing the form.
If the National had truly cared about this issue, then they should have done a risk assessment, and determined that a change in the wording of the form was necessary. If they had, than the rights of association would still apply (v. Dale), since they didn't, they are stuck with what they are stuck with.
Just because we all think we know what a boy is, does not make it true. Changing definitions is what the legal profession lives for.
The National policy change was not one intended to increase membership - no one should really believe that. It was a legal assessment that they determined they would probably lose (since it would now be discrimination of someone who was already a member) or the "cost" of winning (financially or in the public opinion) was not worth it.
It was similar reasoning to the change in adult membership. The BSA "employs" people (i.e. summer camp staff), and must abide by employment laws; being forced to address it in one space, just made it untenable to fight it in the other.
Just because this fire needs a little more gasoline, the full implication is that, legally speaking, your District DE (or camp staff) could be a gay transgendered atheist, and the BSA could not choose to not employ them for only those reasons - regardless of the program the BSA is trying to promote. And I will admit, that even 25+ years ago when I was hiring my camp staff, when interviewing non-scout applicants; I never would have even thought to ask these questions - nor would I have been allowed to even then.
We may or may not like the reality of the times, but, for now, that is the reality we must deal with. We do not do our scouts a service if we hide the realities of the world from them; or spend all our time trying to apologizing to (or gripe at) them that the program they are stuck with is no where near as good as it was when we were boys.