A paradox is essentially, in time travel, is "a situation in which the effect of an incident contradicts or eliminates the cause of that same incident".
There is a paradox in Back to the Future Part III. You want to hear it? I don't care if you don't. Here it is.
In the film, Doc and Marty see Doc's gravestone. It said that on September 7, 1885, Doc was shot in the back by Buford Tannen over a matter of $80. So, 1955 Doc sends Marty to September 2, 1885 to try to get Doc back to 1985. However, the gasoline comes out of the DeLorean, and they can't get the DeLorean to 88 mph, leaving both of them trapped in 1885, where Doc would be shot. Right when Buford is about to kill Doc, Marty intervenes and prevents Buford from killing Doc. And at the end of the film, we even see the gravestone vanish.
And here's our problem:
Marty took a photograph of the gravestone in 1955. If the gravestone vanished from the picture taken in 1955, that means the gravestone never existed in 1955, which means: how could Marty have taken a picture of the gravestone? How could Marty have known Doc would get shot by Buford? And if he didn't, why would Marty go back to 1885 in the first place? And if Marty hadn't gone back to 1885, Buford would have shot Doc. And that means that we should see the gravestone in 1955! Which means Marty should have gone back!
What a paradox.