What is the paradox in "A Sound of Thunder"?

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The time travel paradox is that you might go back in time and meet yourself. 

A paradox is a contradiction.  The problem with time travel is that if it were really possible, nothing could change.  You could always go back in time and reverse whatever happened.  It creates a big mess that gets very confusing and frustrating if you think about it.  That’s why Lesperance tells Eckels that the time machine prevents the paradox, and “When such occasions threaten, Time steps aside.” 

“But if you came back this morning in Time,” said Eckels eagerly, “you must’ve bumped into us, our Safari! How did it turn out? Was it successful? Did all of us get through—alive?”

Travis and Lesperance gave each other a look.

“That’d be a paradox,” said the latter. “Time doesn’t permit that sort of mess—a man meeting himself …” 

That is a handy way to avoid the real problem.  Bradbury is trying to explain that small instances can have huge, life-altering consequences.  When Eckels wants to know how the expedition turned out before it even starts, Lesperance dismisses the idea. 

However, we are also told that even the smallest change in the past affects the future in a big way.  Travis tells Eckels to stay on the path, because the Time Safari only kills animals that are about to die very soon.  That way, there is no alteration. 

“All right,” Travis continued, “say we accidentally kill one mouse here. That means all the future families of this one particular mouse are destroyed, right? 

So, if you step on an insect, you could wipe out a species.  I wonder why the time paradox doesn’t prevent that?  Apparently, it only prevents catastrophic occurrences of a person being in two time periods at once.  The inevitable happens, paradox or not.  Eckels steps on a butterfly and completely changes the future.

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