"Here you can view your collection of plane parts and combine them into planes with the build button"
The Parts screen displays airplane parts that you have purchased at the Market, found at new airports, or generated from scrapping an airplane in the Hangar. There are three types of part available: body, controls, and engine. You are able to sell the parts for the amount of coins listed, gift the parts to a friend via Game Center, or build an airplane from all the parts.
To build an airplane when you have all three parts for that plane, choose the Build button at the bottom left of the screen. Selecting an individual part will bring up the data for that plane, as well as display which parts you have for that plane and how many of those parts you need.
In the game, you can gift parts to your Game Center friends. When you gift a part to a friend, your friend gets the part that you gave them, but this does cost you bux (see this tweet).
Sometimes, during Events there will be 'Mystery Box' jobs. Once you deliver the job, you will get a part of the plane that is being given as a prize for the current global event. Please note that Mystery Part jobs only occur when the prize being offered is a full plane. For example, if the prize was a Bobcat, then the Mystery Box would be a Bobcat part.
A quick way to find Mystery Box jobs is to look at the airport view, and see if a crate is along the back wall of the airport (next to all the other cargo). With this method, you can check more airports, Instead of opening the jobs list at every one.