I have a friend who manages to find free parking near Genthod-Bellevue station, which is 4 stations out of Geneva on the regional train (I might be wrong and it might be Chambesy station, but I think not). That's an easy ride to the airport if you calculate the train times, although you have to change in Geneva Cornavin itself - not difficult.
Otherwise, can a friend not help you out? That's how another friend does it: gets a friend to look after her car!