The Consulate seems to be acting correctly and the parents are misrepresenting it.
They are not, as they put it essentially stateless. They are exactly the same nationalities as always (presumably British) and therefore the DD will be British by descent.
To get a passport overseas, you need the local birth certificate (and possibly a notarised translation).
They do not have and cannot obtain the local birth certificate because of the circumstances of the birth. The consulate has, correctly, advised them that they need a DNA test to establish that the child is theirs (not mistrustful of them per se, but a blanket measure to counter child trafficking). It is up to the parents to arrange this (consulate may be able to signpost suitable providers if they cannot find one)
It's going to be a total PITA for the DC to not have a birth certificate, but a British passport should be an acceptable alternative once back
I have no sympathy for people who give birth overseas who do not make arrangements that cover the time until the baby has the documents needed to travel (unless they travelled very early in pregnancy and got unexpectedly stranded until the birth, in which case they might not have looked in to the formalities beforehand). But if you are planning deliberately to have your child overseas, then you really do need to understand the admin of birth registration at destination and how long it takes to apply for a passport overseas.