Presume you are not staying on a disney resort - cuz then your theme park parking is free. Its just that a lot of people buy an annual pass in order to get a discount on their disney hotel stay- but guess you are not doing this?
Either way - i'd ask anyway at the car park toll, they are usually very accomodating, as long as you can convince them somehow?
I know, wave cash at him - or better still cry - that may work
The other idea about buying in advance at downtown disney is a good one, or even online before you go.