A two-year-old MAY be held on a parent's lap, there is no restriction anywhere, in fact, it is encouraged, that they ride in a safety seat to take advantage of safety belts. When we flew, up to 2-years-old could ride in the lap without paying for a ticket. We bought a ticket to accommodate a safety seat, as our just-under-two-year-old was too big (tall) to hold comfortably on a cross-country flight.
Here is Delta's guideline on that. http://www.delta.com/content/www/en_US/traveling-with-us/special-travel-needs/children.html
If the child is under two, he or she may sit in your lap, but there is no restriction against buying a ticket, or using a ticketed seat for a child under two.
I'm betting the airline wins this one. That seat was NOT for that child.
According to Mr. Schear's logic, he could have put his mother in the seat.
When my son and his wife took their first trip with their son he was 18 months old and it would be roughly 12 hours of flying.
They were going to take turns holding him but I considered that unsafe and said I would PAY for a ticket for him.
They bought the ticket for him themselves,and were glad that they did :-)
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