Transmission problems and rpm on 2010 Chevrolet Cobalt LT?

An auto transmission receives 100% of the motor's drive through a part called the torque converter. This devices basically pushes transmission fluid from a spinning part on the engine side, to the other part that connects to the auto box. If you have no transmission fluid, you have no way to transfer that torque. If you are low, well it makes the job a lot harder. Auto transmissions also have lots of internal piping, when selecting park or drive or reverse, all sorts of valves open and close inside to divert the fluid to different parts of the box. When you stick it in park, it might free up some fluid in the box, that allows more to flow into the torque converter just to get you going.

You say that when you accelerate the rpm's jump up. This very much sounds like low transmission fluid. You need to seriously check the fluid level, and make sure there are no leaks (as it sounds like you have a leak for it to have come out). Check the level, if it's low, add in more. The car might have to be running when you check it, refer to the owners manual for the correct procedure.

The transmission fluid will have two pipes coming out of it going towards the radiator. check those pips for wear, as well as the lower portion of the radiator where the transmission cooler is (most likely part of the engine radiator).