I know something similar to the elephant one (where they don't seem to be related much at all at first!)
There was once a currant bun and a cinnamon roll. On his way out to lunch one day, the roll asked if the bun would like anything to eat. The bun asked for Chinese, but when the roll returned, he had with him pizza. Angry, the bun tossed the roll into the river. What's the moral of the story?
Give the bun what it wants.
And the ever famous one:
How do you put an elephant into a fridge?
Open the door, put in the elephant, and close the door.
How do you put a giraffe into a fridge?
Open the door, take out the elephant, put in the giraffe and close the door.
The lion, King of all the animals, called a meeting. Every animal was expected to attend. Which one didn't, and why?
The giraffe. It was stuck in the fridge.
On the way back to your camp, you have to cross a murky river too wide to jump across. You would swim over, but a sign by the water's edge says: DANGER: DEADLY PIRANHAS AND CROCODILES. There is no bridge, and everything is just flat marshy land for as far as the eye can see. How do you get across without getting bitten?
Just swim across. The piranhas and crocodiles are all at the meeting.
At camp, all you find in your fridge is a giraffe, so you return to the river and go fishing. You feel a tug on the line. What do you reel in?
A cinnamon roll!
Normally the first one can be any kind of a story/joke, as long as something ends up in a river which is not an animal