A joke cited in print from at least 1973 is:
 
Q. What do you get when you cross an elephant with a rhino.
A. An elephino. (Hell-if-I-know—ed.)
 
The joke is also told with RINO (Republican In Name Only). An elephant is a Republican party symbol. “What do you get when you cross an elephant with a RINO?” was told in September 2008; the answer was John McCain, the Republican presidential candidate.
 
       
12 October 1973, The Evening News (Sault Ste. Marie, MI), “Mini Jokes,” The Mini Page, pg. 3, col. 1:
Q. What do you get when you cross an elephant with a rhino.
A. An elephino.
from Jeff Haynes Kincheloe AFB
 
Google Books
That’s so funny I forgot to laugh!:
Jokes, jokes, and more jokes
By Allen Cohen
New York, NY: Dell Pub. Co.
1987
Pg. ?:
What do you get when you cross an elephant with a rhino? Elefino.
 
Google Books
Jokes, Riddles and Wisdom:
The Best Collection Ever
By Roger Kuder
Lincoln, NE: iUniverse.com, Inc.
2000
Pg. 49:
Q: What do you get when you cross an elephant with a rhino?
A: Elephino.
 
LiveJournal 
Eriswarrior wrote,
2006-07-10 07:05:00
(...)
whew, well what do you get when you cross an elephant with a Rino… Eleph-iknow
 
Dammit, Andy!
Monday, September 8, 2008
What do you get when you cross an elephant with a RINO?
(Illustration shows “McCAIN 2008.”—ed.)
As they say, a picture is worth a thousand words.  But I’ll give you a little help understanding this one.  “RINO” is a news junkie acronym meaning “Republican in name only”.  The elephant is the mascot of the Republican party.  Talk amongst yourselves…
Posted by Dammit Andy at 8:18 PM
   
topix (Elizabethtown, KY)
brothafromanotha motha New Haven, KY
Oct 29, 2012
what do you get when you cross an elephant and a rino? elifino
 
The Point Glendale, KY
Oct 29, 2012
brothafromanothamotha wrote:
what do you get when you cross an elephant and a rino? elifino
That was funny when I was a kid, but it’s especially funny now given Republican politics (elephant being the symbol of the party, RINO being the new term for moderate Republicans). I’m going to start saying “elephino” when people ask what party I belong to 😊
   
Before It’s News
Reprising an old joke: What do you get when you cross an elephant and a RINO
Friday, September 27, 2013 10:35
(A cartoon shows establishment Republicans shooting Ted Cruz, a tea party Republican.—ed.)