Are the "extinct species" of meteorites originally from the "Barbarian" asteroids?
The answer is in the part of the abstract that you quote in your other question:


  "...This may be the first documented example of an ‘extinct’ meteorite, that is, a meteorite type that does not fall on Earth today because its parent body has been consumed by collisions."


What they're saying here is that there used to exist some asteroids of some particular type (here they mean formed from the pieces of a single body). After some time all asteroids of that type were destroyed in collisions, fell on planets or into the sun or were ejected from the solar system. Some of the pieces fell on Earth. As far as we can tell there are no pieces left in solar orbit and so none of these type of meteorite fall on Earth any more, hence it is called "extinct".

Since the Barbarian asteroids currently exist, they are by definition not extinct.