Unwarranted claim of higher degree of accuracy in zircon geochronology
I can't be entirely sure but I'll make an informed guess:

That value doesn't come for a single measurement. Therefore, if the error in the age of a single sample is $\pm125$ kyr, you just need to average 16 samples to get it down to $\pm31$ kyr.

The uncertainty in the addition (or substraction) of two or more quantities is equal to the square root of the addition of the squares of the uncertainties of each quantity (assuming they arise from random errors). For example, if we have quantity A with uncertainty $\sigma_a$ and quantity B with uncertainty $\sigma_b$, the error in the quantity $C=A+B$ would be:

$\sigma_c=\sqrt{\sigma_a^2+\sigma_b^2}$

And if we call M to the average between A and B. The uncertainty in the average is

$\sigma_m=\frac{\sqrt{\sigma_a^2+\sigma_b^2}}{2}$

So if we average 16 samples with $\sigma=125$ kyr, the uncertainty in the average would be

$\sigma_m=\frac{\sqrt{16 \sigma^2}}{16}=\frac{\sqrt{16 \times 125^2}}{16}=31$ kyr

Uncertainty propagation can be seen in the abstract of the article you refer to:


  The extinction occurred between 251.941 ± 0.037 and 251.880 ± 0.031
  Mya, an interval of 60 ± 48 ka.


Where the ±48 ka comes from?

$\sqrt{37^2+31^2}=48$

This treatment of uncertainties assumes that uncertainties are independent of each other. As @Mark pointed in the comments, this won't be the case if the uncertainty comes from "the length of your measuring stick (the half-life of U-238)".

This is: if you measure something with a "yard-stick" that have the wrong size, you can't reduce the resulting error by just averaging many measurements.

I don't know enough of geochronology to understand all the different errors they report. But the simple example in the abstract cite I presented above shows that they are indeed treating those errors as independent. Otherwise the reported interval error (±48 ka) would not make sense.

If one significant source of error is indeed the uncertainty in the half-life of $^{238}$U. However, it would be wrong to treat these errors as independent only if there exists an exact value for this half-life. Alternatively, maybe there is no exact value of the half-life, and what they meant is that the half-life value can truly vary a 0.05%. This is something to look into if you want to figure out if the treatment of errors they do is correct. However, after a quick google search I found that radioactive half-life can indeed vary by a small fraction due to environmental conditions, this article explains pretty well the phenomena. Here a short excerpt:


  ...radioactive half-life of an atom can depend on how it is bonded to
  other atoms. Simply by changing the neighboring atoms that are bonded
  to a radioactive isotope, we can change its half-life. However, the
  change in half-life accomplished in this way is typically small. For
  instance, a study performed by B. Wang et al and published in the
  European Physical Journal A was able to measure that the electron
  capture half-life of beryllium-7 was made 0.9% longer by surrounding
  the beryllium atoms with palladium atoms.


If the 0.05% uncertainty on the half-life of $^{238}$U comes from random environmental factors, it would indeed be acceptable to consider them as an independent source of error for each sample.