Can you sue someone for parodying your character? Or is this part of free speech? One thing I am always wondering is why people won't use "Google" in movies, especially cartoons, but parody real people like Donald Trump. It's like you can't hurt the brand of a company, but you can hurt the brand of a person. What does the law say on this in the United States?
A brand name is not a "character". Neither is a real person, unless that person is being used as a character in a work of fiction. I will answer what I think the question intends.
One can't really parody a real person, but one can use a person fictitiously.
I am using here a narrow definition of "parody", that one parodies a work by imitating it so as to comment on it, often but not always negatively. I exclude from "parody": cases where the events of a work are imitated to tell a different story, or the same story in a different setting -- that is "retelling". (West Side Story is a retelling of Romeo and Juliet, not a parody, in this sense.) I also exclude cases where the form or style of a work is imitated to make a different point, or for general humorous effect. That would be satire but not parody.
I am using this narrow sense of parody because it is my understanding that only parody in the narrow sense has been held to be fair use. Satire or retelling is not generally fair use under US law, at least not because of the satire or retelling.
The US Supreme Court has held in Campbell v. Acuff-Rose Music, Inc., 510 U.S. 569 (1994) that a commercial parody can be a fair use. The US Eleventh Circuit court extended this holding to novels in Suntrust Bank v. Houghton Mifflin Co., 268 F.3d 1257 (11th Cir. 2001), in which it held that Alice Randall's 2001 parody, The Wind Done Gone, was not an infringement of Gone with the Wind
But it is important to note that the definition of "parody" used in those cases is significantly more limited than the use sometimes made of the term in artistic circles or by average people.
The courts held that a parody was fair use specifically because it commented on the original. A reworking of a story which did not particularly comment on the original, such as West Side Story based on Romeo and Juliet would not be fair use, at least not because it was a reworking/parody. (Of course Shakespeare is not protected by modern copyright, but if it were, West Side Story might be an infringement.)
The use of the name or description of a real person in a work of fiction would not be an infringement of copyright, as no copyright covers a real person.
Use of a variant of a protected trademark in such a way that reasonable consumers might be confused into thinking that it was created, approved, endorsed or sponsored by the trademark holder would normally be a trademark infringement.
On the other hand a trademark may be used to talk about a product, using the trademark as the name of the product. This is known as nominative use.
Thus a character in a book or movie could say "I bought a Ford" or "I looked it up on Google" and it would not be trademark infringement, usually. However, authors and particularly movie-makers do not want to risk a possibly expensive trademark suit and so they often invent product names to avoid any possible risk of trademark infringement claims. The names of people are less often protected than trademarks, although some are.
Some fictional characters have their names or images protected as trademarks, but most do not. Use of the name of a real person could not be trademark infringement, unless that person's name was a protected trademark (rare but it sometimes happens). Even then, use would probably be nominative use unless the public would be likely to be confused into thinking that the person had approved or endorsed the work.
If a person is used as a character in a work of fiction, and the character is portrayed in a negative way, this might give rise to a claim for defamation.
For that claim to hold up, reasonable people would have to think that the description in the work of fiction also applies to the actual person. The work would have to make false factual statements about the person, that harmed the person's reputation.
If the suit were in the US, and the person defamed was a public official or a public figure, the plaintiff would have to prove actual malice. That means proof that statements of fact were made knowing that they were false, or with reckless indifference to the truth. Proof of actual malice is often hard to obtain.
Note that a work of fiction can be defamatory if a person is readily recognized, even if the name and some details were changed. For example the movie Citizen Kane was clearly based on the life of William Randolph Hearst, and Hearst is said to have considered a defamation action.