First off they don't develop parts overnight, it might even take a whole season to develop a new Rear Wing, for example. Second, you have to assign people to research the parts you want to be developed.
Like this (Design->Manufacturing):

Second, you have to make sure that your personell have sufficient time to develop the parts. Go to R&D->R&D, then click on the clock icons at the bottom of the screen to adjust the time they work with new parts. You should take note that the Designers' "working on the race cars" is actually the time they spend designing your next year's chassis, so don't reduce that time too much. If Engineers spend too much time with R&D, your reliability sucks, whereas too much R&D with Mechanics causes slower part replaceming and car set-upping and more mistakes during pit stops. Adjust them with care.

If you've done it right, the small, white thermometer (the one next to the part names at the Manufacturing screen) should start to turn more and more red until the research is complete. Remember that you can develop a part only once a year, after which you can't develop that part until the next season starts (you can still steal new parts of that type, but you cannot research better ones on your own).