This is not a discussion of parents actions for their children. It is a discussion about eligibility by reason of parentage to be POTUSA. Comparative analogies can be very misleading if not deceptive or even non applicable.
The English language is the same in both instances.
1. "Children born to US citizens abroad are natural born citizens."
2. "Children must be picked up after school by their parents."
In neither instance does the English language say specifically that BOTH parents are necessarily required.
Aside from which, again, Bayard made it clear: Being born a citizen is all that's required in order to be eligible to be President.