  New experiments to measure neutrinos in the far-forward region at the Large
Hadron Collider (LHC) are under design or already in preparation. Two of them,
FASER$\nu$ and SND@LHC, are expected to be active during Run 3 and have the
potential to detect neutrinos that come from high-energy collisions in one of
the LHC interaction points, extracted along the direction tangent to the beam
line. Tau neutrinos and antineutrinos come predominantly from $D_s^\pm$
production in $pp$ collisions, followed by the leptonic decay of these mesons.
Neutrino pseudorapidities in the range of $\eta>6.9$ and $\eta>8.9$ are
relevant to these future experiments. At such pseudorapidities at high
energies, theoretical predictions for the flux of tau neutrinos rely on parton
distribution functions (PDFs) in a combination of very small and large
parton$-x$ values. We evaluate PDF uncertainties in a next-to-leading order
(NLO) QCD calculation of the flux of $\nu_\tau$ + $\bar{\nu}_\tau$ produced by
$D_s^\pm$ decay in the far forward region at the LHC. The theoretical
uncertainty associated with the 40 PDF sets of the PROSA19 group amounts to
$\pm (20-30)$\% for the ($\nu_\tau$ + $\bar{\nu}_\tau$) number of
charged-current (CC) events. Scale uncertainties are much larger, resulting in
a range of CC event predictions from $\sim 70\%$ lower to $\sim 90\%$ higher
than the central prediction. A comparison of the predictions with those
obtained using as input the central PDFs from the 3-flavour NLO PDF sets of the
CT14, ABMP16 and NNPDF3.1 collaborations show that far-forward neutrino energy
distributions vary by as much as a factor of $\sim 2-4$ relative to the PROSA19
predictions at TeV neutrino energies. The Forward Physics Facility in the high
luminosity LHC era will provide data capable of constraining NLO QCD
evaluations with these PDF sets.
