The distances to star-clusters hosting Red Supergiants: $\chi$ Per, NGC
  7419, and Westerlund 1
  Galactic, young massive star clusters are approximately coeval aggregates of
stars, close enough to resolve the individual stars, massive enough to have
produced large numbers of massive stars, and young enough for these stars to be
in a pre-supernova state. As such these objects represent powerful natural
laboratories in which to study the evolution of massive stars. To be used in
this way, it is crucial that accurate and precise distances are known, since
this affects both the inferred luminosities of the cluster members and the age
estimate for the cluster itself. Here we present distance estimates for three
star clusters rich in Red Supergiants ($\chi$ Per, NGC 7419 and Westerlund 1)
based on their average astrometric parallaxes $\bar{\pi}$ in Gaia Data Release
2, where the measurement of $\bar{\pi}$ is obtained from a proper-motion
screened sample of spectroscopically-confirmed cluster members. We determine
distances of $d=2.25^{+0.16}_{-0.14}$kpc, $d=3.00^{+0.35}_{-0.29}$kpc, and
$d=3.87^{+0.95}_{-0.64}$kpc for the three clusters respectively. We find that
the dominant source of error is that in Gaia's zero-point parallax offset
$\pi_{\rm ZP}$, and we argue that more precise distances cannot be determined
without an improved characterization of this quantity.
