“I try to do a bit of exercise every now and then. Last week I was playing Frisbee with my dog, It wasn’t very good. I’ll have to get a flatter dog” is a joke from Irish actor and comedian Joe Rooney that was printed in the Atlanta (GA Journal-Constitution on March 17, 2006. The joke is that the speaker hadn’t been throwing the frisbee to his dog—he’d been throwing the dog, like one would throw a frisbee.
 
Many comedians have told the frisbee joke and authorship is uncertain.
 
 
Wikipedia: Frisbee
A frisbee (pronounced FRIZ-bee, origin of the term dates to 1957, also called a flying disc or simply a disc) is a gliding toy or sporting item that is generally made of injection molded plastic and roughly 8 to 10 inches (20 to 25 cm) in diameter with a pronounced lip. It is used recreationally and competitively for throwing and catching, as in flying disc games. The shape of the disc is an airfoil in cross-section which allows it to fly by generating lift as it moves through the air. Spinning the disc imparts a stabilizing gyroscopic force, allowing it to be both aimed with accuracy and thrown for distance.
         
17 March 2006, Atlanta (GA) Journal-Constitution, “All things Irish,” pg. H3:
The Irish have long been known for their sense of humor. After all, this is the country that gave us limericks. Irish comedian Joe Rooney will perform at “St. Paddy’s Comedy Weekend” at the Funny Farm in Roswell. He gave us this small sample of Irish humor:
(...)
. I try to do a bit of exercise every now and then. Last week I was playing Frisbee with my dog, It wasn’t very good. I’ll have to get a flatter dog.
     
28 December 2008, Mail on Sunday (London, UK), “We are the Pantocracy!” by Gwyneth Rees, pg. 28:
BOBBY DAVRO, 50
Who: Comedian and impressionist. Performing in: Cinderella, Bristol Hippodrome. Playing: Buttons. Funniest line: ‘I took my dog for a walk on Christmas Day and I started playing frisbee with him but he was rubbish. I need to get a flatter dog.’
 
Google Books
Two Nuns In A Bath
By Geoff Tibballs
London, UK: Constable & Robinson
2009
Pg. ?:
A man said to his friend: “I took our dog for a walk the other day and played Frisbee with him, but he was useless. I really need to get a flatter dog.”
   
Twitter 
Tim O’Neill
@dezignguy
I took our dog for a walk the other day and played Frisbee with her, but she was useless . I really need to get a flatter dog.
9:42 AM · Apr 5, 2010·Twitter Web Client
   
Twitter
Rick Sadecky
@timeclockman
I tried to play Frisbee with my dog, but it was no use. I need to get a flatter dog.
9:10 PM · Apr 12, 2010·Twitter Web Client
 
Google Books
Jokes for Blokes:
The Ultimate Book of Jokes Not Suitable for Mixed Company
By Llewellyn Dowd and Phil McCracken
London, UK: Ebury Press
2011
Pg. 162:
I went to the park this morning to play frisbee with my dog. He was useless. I think I need a flatter dog.
     
8 March 2012, The Daily Mirror (London, UK), “Bet of the Day” by Derek McGovern and John Shaw, pg. 56:
I TOOK my dog to the park and played Frisbee with him. It was a complete waste of time - I think I need a flatter dog.
 
Twitter
Shit Jokes
@ShitJokes
Went to the park today to play frisbee with my dog but it was no fun.
I need a flatter dog
3:24 PM · Oct 29, 2015·Twitter for Android
 
Twitter
🤣 The Dad Joke Man 😉
@DadJokeMan
Played frisbee in the park with my dog yesterday. Not much good though, I think I’m going to need a flatter dog….              #fridayfunny
2:02 AM · Oct 5, 2018·Twitter for iPhone
 
Reddit—Jokes
Posted by u/man_nowhere February 23, 2019
Went to the park yesterday to play frisbee with my dog.
Think I’m gonna need a flatter dog.