Aside from incorporating a new multi storey within airport city, the airport own & operate many car parks around the airport. Almost any of these could be converted into multi storey parks.
The advantages of doing this are three fold, 1) less environmental damage concreting over green fields, 2) remaining a good neighbour to local residents & 3) retain a safer clear area under the flight path.
The airport appears to have chosen the cheapest & most disruptive option.
The residents have seen the airport grow & unlike Heathrow, have movements H24, then they got the train,now the trams will soon be thundering past their houses & now a 9,000 space car park on their doorstep. It was an avoidable decision, when there are clearly alternatives available with no detriment to the airport expansion.