For most atheists, no.

Like the question of an afterlife or a god, we're waiting on good evidence that a soul or spirit exists.

Atheists tend to believe that the mind is an "emergent property" of the brain, in the same sense that electrical currents within a computer's circuitry manifest as a webpage loading into a browser. We don't currently have any examples, at all, of a mind that exists outside of a physical medium, such as a computer or biological brain.

As far as we can tell, the mind is essentially software, and it doesn't have any existence in parallel dimensions, that we know of.