Well, if you select a mission in the quick-mission-menu, you will see the planes for both sides (in some missions you got several plane-types per side). Above the plane-pictures, you will see pilot-faces/bodys. By klicking on those pilote-faces, you select your plane. The ones you did not select, are more transparent.
In this picture, the hurricane is selected:
http://www.simhq.com/_air15/images/air_547a_008.jpg
But attention: You are only able to select flyable aircraft. So if klicking on it does not work, the plane is probably not flyable.