Efrem

I agree that they're unlikely to notice. The people who check IDs against tickets will not see a DOB on the ticket. The only people who have access to both pieces of information are airline gate or check-in agents. They are unlikely to compare them and even less likely to care if they don't match.

However, in the very unlikely event that one of them says something, since they're both there, they only have to explain that the person who booked their tickets reversed the birth dates. The agent can easily confirm this. Again, no problem.

The only time there will be a problem is if someone known to TSA as a potential troublemaker has the name of one of your parents and the birth date of the other. While I don't know any of those four data items and have no access to TSA databases, I suspect we can safely ignore that possibility.