Some of the Doctor's statements in "The Stolen Earth" seem to imply that, after the Last Great Time War, he's the only Time Lord left in any reality. The fact that there were alternate versions of almost everyone but the Doctor in "Father's Day" might also support that.
Of course even if this is true, that doesn't tell us whether there were other Gallifreys, other Time Lords, and even other Doctors before the War.
In the classic series, "Inferno" was originally going to feature an alternate Third Doctor, but the scenes were cut from the final version. Later, an early draft of the "Battlefield" script featured Merlin as an alternate Seventh Doctor, but this was changed to instead imply that Merlin (who was never seen) was a future incarnation, or possibly even our universe's Seventh Doctor meeting Arthur's knights out of order.
So, the TV show never established the answer. The novels later directly confirmed that at least some parallel universes had their own versions of the Doctor—for example, in Blood Heat, the Seventh Doctor found the corpse of the Silurian World's Third Doctor and took his TARDIS.