This expression is sometimes used to express that you can trust what a person (or, in the last example below, a puppy) says. Here are a few examples:

From http://blog.bellatheboxer.com/guidelines-and-disclosures.htm... :
"Your privacy is very important. As such, your personal contact information will never be shared with anyone. Dog’s honor!"

From http://www.drdogsbehaviorsolutions.com/index.php :
"I will see your name and email address and promise to never sell or share them with anyone. Dog’s honor!"

From the comments at http://www.fififlowers.com/2008/09/location-location-locatio... :
"Alfie promises to do a good job and be a great listener today for you Fifi. Dog's honor!"

From the "Pee-Mail" sidebar at http://mrappgate.blogspot.fr/2011/12/good-day.html :
"Sometimes Mommy hogs the computer though and forgets to let me check it, so I may not pee back for a little while. I will though. Puppy dog's honor!"

It seems pretty close to the original text.

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Note added at 8 hrs (2012-12-09 09:25:03 GMT)
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Here are some more examples, this time with the British spelling:

From http://www.sofurry.com/view/256648/guest :
"Dutifully, Fritz stopped several paces away, flopping down on his side submissively, exposing himself if the umbreon chose to attack. Not usually wise in the presence of a strange canine, but this one would certainly expect the extra courtesy and the appearance of complete capitulation. "Like I said man, I'm just here for the action, ya know? You won't get any trouble from me, dog's honour." He raised a paw, flashing a grin that showed all his teeth."

From a transcript of "The Garfield Show," as said by Odie the dog, at http://www.livedash.com/transcript/the_garfield_show-(orange...
"00:11:46 >> Wait!
00:11:47 How do we know you're telling the truth?
00:11:50 >> I swear.
00:11:51 Dog's honour!"

From a blog by a dog at http://emberdog.com/diary_08_sep_2.html
"I've got a suspicion that we might be going home tomorrow, as bags have been moved about. I'll be sorry if we have to leave so soon, as I'm enjoying Cornwall very much.
I hope you like the picture of the secret door. I promise I'm not making it up. Dog's honour. Woofs and licks to all."

And finally, here's one from a comment on an aggregator of children's publishing blogs, at http://www.jacketflap.com/megablog/?highlight=The Sense of H... :
"As a dog lover I enjoyed this. Funny, but full of good advice. I'm in the process of designing my own website - I know, I know, I can hear your shout of horror from here - but I really can't afford to have it done professionally. I'm off over to yours to check it out for ideas (no stealing, dog's honour)!"

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Note added at 11 hrs (2012-12-09 12:46:56 GMT)
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I think it's quite possible that "parole de toutou" may be intended to evoke "parole de scout" (as a kind of assonance).

I've definitely heard "scout's honor" more than "dog's honor." However, I find the fact that "dog's honor" seems to have been used in a cartoon for children (i.e., in the Garfield transcript link above) pretty persuasive. It's a similar register.