Second, this is carbon dating, and we are talking about drama involving a couple decades.
It is plain that between the actual science and the reporting, there is a weak link (or several), in the reporters.
TFS gives an age range, which if not qualified is probably as you say, a 95% confidence interval. That alone makes it a non-story, really. Their C.I. for the age of the parchment goes before the (nominal) date of birth of the purported author. Non-issue on the face of it, even before you bring up issues like palimpsests.
If I wanted to, and given a reasonable amount of funds, I could start to make parchment tomorrow which would, when prepared, give a carbon date of, say, 200AD. If you used similar techniques to prepare a suitable ink
The technique for preparing the parchment would be to grow a lot of grain in a greenhouse using CO2 from coal manufacture, combined with natural ventilation. The coal-derived CO2 would dilute the natural C-14, as it has essentially no C-14. We want 1400 years of depletion, so that would be about 1/4 coal-derived CO2 to 3/4 natural ventilation. (We might need to filter for N-14 and nuclear-test derived nucleotides, so it might be easier to go with straight coal-derived CO2 and add a source of C-14.)
C-14 depleted grain in hand, we grow a small herd of C-14 depleted sheep or goats. We'd need to check they're genetically similar to ones in the area. Just buy some local goats and we can breed up our beasts.
How to make the ink
And I've got other things that I've got to do. Is the general plan clear? Are you up for it?