Pepe the Frog's Creator Is Sending Takedown Notices To Far-Right Sites (vice.com)
An anonymous reader quotes a report from Motherboard: Pepe the Frog creator Matt Furie has made good on his threat to "aggressively enforce his intellectual property." The artist's lawyers have taken legal action against the alt-right. They have served cease and desist orders to several alt-right personalities and websites including Richard Spencer, Mike Cernovich, and the r/the_Donald subreddit. In addition, they have issued Digital Millennium Copyright Act takedown requests to Reddit and Amazon, notifying them that use of Pepe by the alt-right on their platforms is copyright infringement. The message is to the alt-right is clear -- stop using Pepe the Frog or prepare for legal consequences. Furie originally created Pepe as a non-political character for his Boy's Club comic, but Pepe later became an internet meme and during the 2016 U.S. presidential election the alt-right movement appropriated the frog in various grotesque and hateful memes.
you're thinking of Trademark, this is copyright. He can grant license to and take license from pretty much anyone he damn well pleases. The rules are a little hazy for music because of radio, but print media's pretty cut & dry.
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Again, that is trademark. Copyright has no such requirement.
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Also, in nearly all of the renditions I've seen, it's been used in a satirical sense, mostly to poke fun at anti-fascists, and, even more hilariously, at fascists themselves.
Ironically, your post includes the answer to why this is irrelevant. Notice how your subject line was "parody", but in the body, you say "satire" (well, "satirical")? Those are different things under copyright law.
The short version is that:
(i) parody makes fun of the thing it's copying. Think Weird Al's "Smells like Nirvana", which explicitly makes fun of Nirvana and Smells Like Teen Spirit, or his "Perform That Way" which makes fun of Lady Gaga and Born that Way. Parody falls under fair use because, since you're making fun of the thing you're copying, there's no way to do so without copying it.
(ii) satire makes fun of something else.Think Weird Al's "Eat it" or "I'm fat", which make fun of obesity, but do not make fun of Michael Jackson or those songs, except stylistically. He could have made fun of obesity with countless other songs, so the copyright on those songs do not limit his expressive rights. That's why satire does not fall under fair use.
So, if those renditions you've seen are making fun of, say anti-fascists or Hillary Clinton or what not, they're satires. They are not parodies of Pepe the Frog, and therefore are not protected by fair use, unlike if they had actually been parodies.
As an aside, Weird Al always gets permission from artists before he copies their songs, and while it's primarily because he's such a nice guy, the above satire/parody divide is another significant reason.
Disclaimer: I am an IP lawyer. I am not your IP lawyer, and this is not legal advice.