They have contacted TUI for an explanation as to why the family were left sitting on the floor.
Paula Taylor was travelling to Birmingham Airport with her husband and daughter on TUI Airways.
Paula Taylor told the show that she, her husband and 10-year-old daughter had got to the airport early, in June, to make sure they were seated together.
"We went straight to the front and we were very excited the fact we had managed to sit together".
However, once the plane was in the air, the attendants needed access to the food and duty-free goods which were stored behind those seats. The crew said that for take-off and landing Brooke would take the last spare seat while Paula and Ian, 55, would sit in two spare flip-up "jump" seats in the crew station.
Paula Taylor added, however, that the co-pilot had come to sit with them on the floor, thanking the family for their "co-operation and understanding".
But despite the efforts of staff to minimize their discomfort, the family still spent the flight on the floor.
TUI told the BBC it had offered the family a $1,667 refund.
A spokesman for TUI said: "We are sorry to hear about Mr Taylor and his family's experience with us". The company claimed that the incident was caused by a "last minute aircraft change".
The episode of Rip-Off Britain: Holidays will be broadcast on BBC One on Tuesday at 9.15am.
However she received a full refund after contacting BBC's Rip Off Britain: Holiday.
Under Civil Aviation Authority rules, flyers are not permitted to be left unseated at any stage of a flight. The newspaper explained that passengers must not be without a seat during any point of a flight, though they can use cabin crew seating in some situations.
Mrs Taylor said TUI offered her a £30 goodwill gesture when she raised a complaint.