You have every legal and moral right to create a parody of a work in the US (I know nothing about the situation in the UK), but that does not mean that someone can't sue you.
There's a couple of things to note here.
For a start, yes - there is an exception in copyright law for parodies. However:
1. 'Parody' in copyright law has a very specific meaning and may not line up 100% with the way non-lawyers use the word. It must be about commenting on the copyrighted work - not about commenting on something else.
2. That exception is all about copyright violations - it says absolutely nothing about the thousands of other laws (right of publicity, moral rights etc)
For it to be a 'parody' you must be parodying the actual copyrighted work itself. For example - if I do a parody of 'Lord of the Rings' movie, then I can use the copyrighted script as a basis - because that it what I am parodying. However - that doesn't give an exemption to breaching the copyright of other things - such as the music score - which may not be what I'm actually parodying.
Also it needs to not compete in the same market as the original. (Yes - I'm oversimplifying the heck out of this) So I parody of a drama would become a comedy .. and be a simpler thing legally than a parody of a comedy - which would still be a comedy.
Sometimes people just change the genre of the film and call it a 'parody' to say
"Look - I know I can't legally do this. But I can make you spend a fortune in legal bills to prove it and I don't have many assets so you'll never make a cent back." The porn versions of popular films rely on this - but sometimes the original filmmakers are stubborn enough that they can stop them
People threaten over them all the time .. but often just to the extent of sending a letter or muttering in the press. So the parody 'The Six Thousand Dollar man' (a version of the 'Six Million Dollar Man' during government cutbacks .. so his 'super' strengths were lame) was threatened by .. ironically .. the film makers behind the 'Scary Movie' franchise! Yes, they understood entirely about the parody exception .. but it is sometimes just about stamping your foot and objecting for the sake of it.
They are my thoughts, anyway.
Good luck!
Mac