In the United States, a website that is labeled as parody and that a reasonable reader would, upon reflection, recognize as not stating true facts is protected under the First Amendment, so no liability will attach.
The key case on this point is Hustler Magazine v. Falwell, 485 U.S. 46 (1988). At the time, Campari Liqueur was running an ad campaign in which people talked about their first time drinking Campari, but in a context setting up extended double-entendres about losing their virginity. Playing off that format, Hustler ran a fake interview with Jerry Falwell about his "first time," which can really only be understood if reprinted in full:
Falwell: My first time was in an outhouse outside Lynchburg, Virginia.
Interviewer: Wasn’t it a little cramped?
Falwell: Not after I kicked the goat out.
Interviewer: I see. You must tell me all about it.
Falwell: I never really expected to make it with Mom, but then after she showed all the other guys in town such a good time, I figured, "What the hell!"
Interviewer: But your Mom? Isn’t that a little odd?
Falwell: I don’t think so. Looks don’t mean that much to me in a woman.
Interviewer: Go on.
Falwell: Well, we were drunk off our God-fearing asses on Campari, ginger ale and soda—that’s called a Fire and Brimstone—at the time. And Mom looked better than a Baptist whore with a $100 donation.
Interviewer: Campari in the crapper with Mom. How interesting. Well, how was it?
Falwell: The Campari was great but Mom passed out before I could come.
Interviewer: Did you ever try it again?
Falwell: Sure. Lots of times. But not in the outhouse. Between Mom and the shit, the flies were too much to bear.
Interviewer: We meant the Campari.
Falwell: Oh, yeah, I always get sloshed before I go to the pulpit. You don’t think I could lay down all that bullshit sober do you?
Campari, like all liquor, was made to mix you up. It's a light, 48-proof, refreshing spirit, just mild enough to make you drink too much before you know you're schnockered. For your first time, mix it with orange juice. Or maybe some white wine. Then you won't remember anything the next morning. Campari. The mixable that smarts.
Predictably, Falwell sued, claiming defamation and extreme emotional distress. A jury awarded Falwell $150,000, and Hustler appealed. The Supreme Court reversed, holding that Falwell could not recover for defamation because the ad "was not reasonably believable," and that he could not recover for emotional distress because the First Amendment does permit such claims when "the conduct in question is the publication of a caricature such as the ad parody involved here":
Despite their sometimes caustic nature, from the early cartoon portraying George Washington as an ass down to the present day, graphic depictions and satirical cartoons have played a prominent role in public and political debate. ... From the viewpoint of history it is clear that our political discourse would have been considerably poorer without them. ... There is no doubt that the caricature of respondent and his mother published in Hustler is at best a distant cousin of the political cartoons described above, and a rather poor relation at that. If it were possible by laying down a principled standard to separate the one from the other, public discourse would probably suffer little or no harm. But we doubt that there is any such standard, and we are quite sure that the pejorative description "outrageous" does not supply one.
Since then, the principles from Hustler have evolved and distilled into a test that generally requires the plaintiff to prove that reasoanble readers would view the material in question as actually telling the truth about the defendant. For your hypothetical, then, the question would become whether people reading the website believed that about it.
This is a trickier question than it might sound like. For more elaboration on the boundaries of parody and how we identify, you could check out the decision in Novak v. City of Parma, 932 F.3d 421, 427–28 (6th Cir. 2019).