When was the distance to a star measured for the first time without using parallax? This is somewhat of a follow-up to When was the parallax of a star first measured? Once the distance to the nearest stars was determined, it was possible to discover physical properties of stars (such as the relation between intrinsic luminosity and color, or the relation between luminosity and period for Cepheid variables), then use the knowledge of those physical properties to find the distance of stars with no visible parallax.
When was the distance to a star (other than the Sun) determined for the first time with a method other than parallax?
From Wikipedia

In his 1698 book, Cosmotheoros, Christiaan Huygens estimated the distance to Sirius at 27664 times the distance from the Earth to the Sun (about 0.437 light years, translating to a parallax of roughly 7.5 arcseconds).

and

It was also in this book that Huygens published his method for estimating stellar distances. He made a series of smaller holes in a screen facing the Sun, until he estimated the light was of the same intensity as that of the star Sirius. He then calculated that the angle of this hole was {\displaystyle 1/27,664}1/27,664th the diameter of the Sun, and thus it was about 30,000 times as far away, on the (incorrect) assumption that Sirius is as luminous as the Sun. The subject of photometry remained in its infancy until the time of Pierre Bouguer and Johann Heinrich Lambert.[124]