If you have the game sounds turned on before a stork enters the screen you will hear a baby cry. The stork can be collected with or without a parachute and it will add a life that has been lost.
Most of the time if a storm cloud is on screen a stork is also arriving.
There is no screen in the game which tells you how many storks you have collected.
Note from Prince Keepo: If you're collecting storks, then always have at least one life below the max amount and they will appear more frequently.