nick-- you can give questions stars but you can only give a thumbs up or thumbs down to answers.
The parallax formulas are used for stars up to 100 million light years away. For stars beyond 100 million light years other systems are used. Your question lies on the border. As parallax has already been explained, I offer complete measurement systems for stars beyond 100 mly as some answers have lacked.
The methods of measuring distance to stars beyond 100 light-years:
Cepheid variable stars. These stars change in brightness over time, which allows astronomers to figure out the true brightness. Comparing the apparent brightness of the star to the true brightness allows the astronomer to calculate the distance to the star.
Stellar motions: All stars are in motion, but only for nearby stars are these motions perceivable. Statistically, therefore, the stars that have larger motions are nearer. By measuring the motions of a large number of stars, we can estimate their average distance from their average motion.
Moving clusters: Clusters of stars travel together, such as the Pleiades or Hyades star clusters. Analyzing the apparent motion of the cluster can give us the distance to it.
Inverse-square law: The apparent brightness of a star depends both on its intrinsic brightness (its luminosity, or how bright it really is) and its distance from us. If we know the luminosity of a star (for instance, we have a measured parallax for one star of the same type and know that others of the same type will have similar luminosities), we can measure its apparent brightness (also called its apparent magnitude) and work out the distance using the inverse-square law. There are several variations on this, many of which are used to measure distances to stars in other galaxies.
Interstellar lines: The space between stars is not empty, but contains a sparse distribution of gas. Some times this leaves absorption lines in the spectrum we observe from stars beyond the interstellar gas. The further a star is, the more absorption will be observed since the light has passed through more of the interstellar medium.
Period-luminosity relation: Some stars are regular pulsators. The physics of their pulsations is such that the period of one oscillation is related to the luminosity of the star. If we measure the period of such a star, we calculate its luminosity. From this, and its apparent magnitude, we can calculate the distance.