Copyright governs who has the "right" to produce and distribute "copies" of books/music/movies/creative works. This is where fair use doctrine applies, because most creative works are referential by nature.
Weird Al is allowed to parody everything because he's operating under copyright law, not trademark law.
Trademark governs who can "trade" under what "mark" i.e. the brand identity of a company. Companies don't own their trademarked word forever, but they maintain the exclusive right to sell things under that brand in their specific market sector. Patagonia doesn't own the name of a geographical region, they just own the right to be the only company using that name to sell clothing and outdoor gear.
A drag queen name can be a parody of a clothing and outdoor gear company.
A company's trademarked logo can be used in parody creative works, with more leeway if it's not for commercial purposes. Trademark parody is allowed! Patagonia has been aware of and allowed Pattie Gonia's trademark parody for years.
Trademarks are specific to market sector. Actress Chase Infiniti could start a makeup line named after herself and her trademark would not infringe on the Infiniti car brand because they are different markets and there is no risk of confusion. Pattie Gonia could probably trademark her name to sell frozen veggie burgers and Patagonia would not care.
Drag queen Jan Sport did a collab with JanSport bags. What Jan Sport almost certainly did not do is independently apply to register "Jan Sport" as a trademark in order to sell bags on her own, because that would infringe on JanSport's own trademark in the bag market sector.
What Pattie Gonia is not allowed to do -- the thing that Pattie Gonia actually did do and is being sued for -- is apply to register "Pattie Gonia" as a trademark to sell clothing, because apparently Pattie is in talks with North Face and HydroFlask to sell "Pattie Gonia"-branded gear. These companies probably won't finalize anything unless Pattie shows that she actually owns the trademark. Unfortunately, "Patagonia" is already a registered trademark in the clothing market sector, and these two names are too similar to exist in the same sector (see: "likelihood of confusion" legal standard).
Your drag queen name can parody a clothing company. You can parody the trademarked logo of a clothing company. But you cannot use the same name to then go on to also become a clothing company.
In order to maintain their own trademark, Patagonia must sue for trademark infringement. If they don't sue, and Pattie Gonia gets her own trademark, Pattie could sue Patagonia for infringement on her trademark. You can see why Patagonia won't be dropping this suit no matter how much you harass them.
Yes, Pattie's legal fees to fight this will cost more than the $1 she's being sued for. Pattie could also not fight this, withdraw her trademark application, not spend any money, and carry on being an environmental activist drag queen named Pattie Gonia. She would probably be better off making nice with Patagonia in the hopes of a Jan Sport-esque deal where Pattie designs an exclusive fabric and Patagonia maintains the trademark, but apparently Pattie's legal team has been sassing off to Patagonia in their communications for years, has applied for a trademark they should 100% know they'll never get, and has now decided to play the victim on social media just in time for Pride month, so I don't know how likely that is. I guess we'll see!
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