OP paid for the two FC tickets with Avios (BA or IB????) and only the three teens in coach with AA miles. It's not clear that anyone contacted AA. It's also not clear to me why the three teens could not have flown to Spain and gone to the hotel on their own with parents to follow the next day, or did IB deny them boarding too? Otherwise, a kid of 19 should be able to manage this while the others age 17 and 15 are too old for UM services to be required or even permitted. Were the three kids also denied boarding or was it the parents decision that they would not fly alone, on a separate PNR? Were the PNRs linked?

It sounds like both the USA DOT and EC 261 IDB regulations would apply, although I wonder whether the flight was oversold in FC or just in coach.

At any point in the booking process or on the receipt/itinerary, was the need for the credit card indicated? What about in the IB (and possibly BA if the ticket was booked with BA Avios) CoC and FF program rules?

Did OP do OLCI for parents and/or kids? If so, was a need for the credit card indicated at that time? Was anything said at the counter when the family checked in or checked bags or did the problem truly arise only at the GATE?