TUI told the BBC it had offered the family a $1,667 refund. Mrs Taylor told the BBC: 'It's hard and its uncomfortable and it's just filthy.
However, Mrs Taylor said that she "got short shrift" from the airline when she personally complained immediately after the incident, despite having photographs of what had happened.
Under Civil Aviation Authority rules, flyers are not permitted to be left unseated at any stage of a flight.
'We just couldn't believe it when we realised that there genuinely weren't any seats for us'.
But despite being given seats 41 D, E and F, when they boarded the plane they found only empty spaces where their chairs should have been.
Once all the passengers had boarded there was just one seat left. Though they agreed, Paula and Ian were soon asked to move to the floor so staff could access the food and duty-free items stored nearby.
When the plane took off, Taylor's 10-year-old daughter took the one seat that was available on the plane, and her parents sat in crew seats.
The couple were joined by Brooke as she did not want to sit alone.
The family say they were thanked by the plane crew for their understanding.
Tui said the family were offered the full £1,300 of the flight to be refunded and £30 as a "goodwill gesture".
It said it was "sorry for the way the situation was initially handled" and will contact the family directly to apologise.
The company has been contacted for further comment by BBC Online.
Paula Taylor was travelling to Birmingham Airport with her husband and daughter on TUI Airways. Frustrated, she contacted Rip-Off Britain.