The person whose name was tied to that seat did not board the plane. The parents bought the seat but the person that was assigned to that seat left on an earlier flight so the parents decided to use it for their two-year-old that did not have a boarding pass or seat assignment.
Nobody checked in for that assigned seat. It at that point becomes vacant and available for other passengers that want to take the flight.
You can't buy two seats just to have an extra next to you apparently is the rule...unless you are too fat and the make you buy two seats.
I'm saying the parents were not being reasonable and were not following the rules. I side with the airline on this one.