What am I missing when blowing out water from my sprinkler system?
My guess would be that there is something wrong with the backflow preventer. 


Most likely, the air inlet valve is not sealing properly. When pressurized from the side marked "normal flow" (which is where your tap is), it should open the check valve, and also force the air inlet valve shut. 

It's probably a rubber gasket, and might be worn out and need to be replaced, or maybe something is physically holding it open. Try pushing on it to see if you can get it to close all the way. Some silicon grease around the gasket will help it form a good seal and prolong the life of the gasket.