Of course, dogs can’t be licensed to practise medicine either, but catscan.
You win the thread.
A woman brought a very limp parrot into a veterinary hospital. As she lay her pet on the table, the vet pulled out his stethoscope and listened to the bird’s chest. After a moment or two, the Vet shook his head sadly and said, “I’m so sorry, Polly has passed away.”
The distressed owner wailed, “Are you sure? I mean, you haven’t done any testing on him or anything. He might just be in a coma or something.”
The vet rolled his eyes, shrugged, turned and left the room returning a few moments later with a beautiful black Labrador. As the bird’s owner looked on in amazement, the dog stood on his hind legs, put his front paws on the examination table and sniffed the dead parrot from top to bottom. He then looked at the vet with sad eyes and shook his head.
The vet led the dog out but returned a few moments later with a cat. The cat jumped up and also sniffed delicately at the ex-bird. The cat sat back, shook its head, meowed and ran out of the room. The vet looked at the woman and said, “I’m sorry; but like I said, your parrot is most definitely, 100% certifiably …dead.”
He then turned to his computer terminal, hit a few keys and produced a bill which he handed to the woman. The parrot’s owner, still in shock, took the bill. “$150!” she cried. “$150 just to tell me my bird is dead?!”
The vet shrugged. “If you’d taken my word for it, the bill would only have been $20, but with the Lab Report and the Cat Scan, what did you expect?”
The joke is actually a really nifty example of zeugma
A customer enters a pet shop.
Mr. Praline: 'Ello, I wish to register a complaint.
…
It’s in case they get “carded”. Because dogs are “social animals”, so they like to go out drinking together. 
[Moderating]
Snake_Plissken, I know that it’s commonplace to quote a few lines of Monty Python here and there, but transcribing, verbatim, an entire routine goes well beyond that, into the realm of copyright violation. We’re pretty strict about copyright, around here, and so I have edited your post (it was the Dead Parrot skit, in case it wasn’t obvious).
There’s a whole series, too, of complete non-sequitur jokes. The one I remember is:
Q: Why is a fish like a tomato?
A: Neither can ride a bicycle.
I suppose it falls in the category of so absurd and illogical that it’s funny.
What’s brown and sticky? a stick. Or less absurd - why do firemen wear red suspenders? To hold up their pants.
I think total 180-degree turns in language and/or logic appeal to kids who are encountering them for the first time.
Recall the story I’m sure everyone’s heard about the guy in the bar who for $100 apiece will show his dog can talk. A few guys take the offer.
He asks the dog “how do you describe sandpaper?” the dog answers “Rough!”
He asks “what’s on top of a house?” and the dog says “Roof!”
He asks “Who was the greatest baseball player of all time?” and the dog say “Ruth!”
The guys in the bar don’t find this funny, beat the crap out of the guy and toss him out.
The dog looks at him and says “Should I have answered DiMaggio?”
I think it’s because “license” by itself is usually assumed to mean “driver’s license.” It turns out to be a swerve because this type of license is a dog license.
No, because that wouldn’t be funny. The whole point of the humour derives from the silliness of the concept that cats can’t drive, but dogs can.
Another example - -
Two goldfish are in a tank.
One turns to the other and says “do you know how to drive this thing?!”
If it’s a kid’s joke, the silliness is the important part of it. Not only doesn’t it have to make logical sense, it shouldn’t.
What’s the difference between an elephant and a grape?
Grapes are purple.
What did Tarzan say when he saw the elephants come over the hill?
Here come the elephants over the hill!
What did Jane say when she saw the elephants come over the hill?
Here come the grapes! (She was colorblind.)
There are non sequitur jokes, and there are (for want of a better term) super sequitur jokes. Colibri’s joke about what Tarzan said, and mine about what the farmer said, are both super sequiturs. The humor comes from the punchline being so unsurprising that it’s surprising. In a way, it’s similar to some shaggy dog stories, except that it’s short.
The old “why did the chicken cross the road” joke is a kind of super sequitur. The joke is so old, and so often repeated, that we don’t usually think of what’s supposed to be funny about it. The humor is in the how mundane the answer is.
Nah, it still works as humor told the way cmkeller tells it. It’s the redirect of the meaning of the word “license” that gives it that groaning chuckle. Typical kids joke.
It is more absurd as told in the McDonald’s activity pack, though.
Speaking of dogs driving, one of my favorite ads of all time:
Q: What do you call a boomerang that doesn’t return?
A: A stick.
Q: Where can you find a llama with no legs?
A: Wherever you left him.
Then there’s the reason my wife tells me I’m not allowed to tell jokes to children:
“Mommy, mommy - I hate my sister’s guts!”
“Shut up and eat what you’re given.”
Not much beats a good dog joke!
Policeman: “Excuse me Mr, but were you aware that your dog has been chasing a guy on his bike”
Dog Owner: "Are you nuts? My dog is not even able to ride a bike
Slightly zombie thread I know, but there also used to be a joke about buying a black and white dog because the licence is (was) cheaper.
(it wasn’t, but a TV licence was cheaper if you only had a black and white set)