The starmap of Hipparchus established the coordinates for about 850 stars. His map did not include parallax measurements This would not begin until 1840 when parallax was measured for three stars: 61 Cygni by F. W. Bessel, Vega by Wilhelm Struve, and Alpha Centauri by Thomas Henderson. The significance of having parallax measurements was not merely to distinguish these stars from the background stars. Halley had already done this in 1718 by identifying three stars ( Sirius, Procyon, and Arcturus) that had moved in relation to background stars. A parallax measurement establishes the distance to the star. By 1900 about 90 of the brightest stars had parallax measurements and therefore known distances.

I have not yet found a source for parallax advancements for the early 20th century so let me jump ahead. The Hipparcos Catalog, a high-precision catalog of more than 118,200 stars, was published in 1997. The lower-precision Tycho Catalogue of more than a million stars was published at the same time, while the enhanced Tycho-2 Catalogue of 2.5 million stars was published in 2000. Hipparcos‘ follow-up mission, Gaia, was launched in 2013. Gaia increases the number of stars with parallax measurements to a billion.

With this great progress, it might seem that the reach was also immense. However, the reach of parallax is limited to our galaxy, the Milky Way, and not the entire galaxy. Figure 1. illustrates the reach.

 Figure 1. The Limits of Parallax                                                     ESA/Hubble

The left half of Figure 1 shows the bottom right quarter of the Milky Way galaxy. The largest dashed circle shows the current domain for parallax.

The Gaia Project collects among other things parallax measurements, repeated coordinates of position, luminosity, and color. See this video from the first release of data which shows the motion of two million stars of the Milky Way Galaxy projected for the next five million years. 
https://www.youtube.com/watch?v=Ag0qsSFJBAk