Telescopes were apparently invented in 1609, but didn't become advanced enough to measure stellar parallax until the 1830s.
Observation of stellar parallax would be a big step in proving the heliocentric theory, and I think that the lack of detectable stellar parallax was used as an argument against the heliocentric theory in ancient times. It was certainly used as an argument against the heliocentric theory in early modern times.
Stellar parallax is so small that it was unobservable until the 19th century, and its apparent absence was used as a scientific argument against heliocentrism during the early modern age. It is clear from Euclid's geometry that the effect would be undetectable if the stars were far enough away, but for various reasons, such gigantic distances involved seemed entirely implausible: it was one of Tycho Brahe's principal objections to Copernican heliocentrism that for it to be compatible with the lack of observable stellar parallax, there would have to be an enormous and unlikely void between the orbit of Saturn and the eighth sphere (the fixed stars).1
After the Copernican theory gained popularity astronomers made many attempts to measure stellar parallax.
In astronomy, aberration (also referred to as astronomical aberration, stellar aberration, or velocity aberration) is a phenomenon which produces an apparent motion of celestial objects about their true positions, dependent on the velocity of the observer. It causes objects to appear to be displaced towards the direction of motion of the observer compared to when the observer is stationary. The change in angle is of the order of v/c where c is the speed of light and v the velocity of the observer. In the case of "stellar" or "annual" aberration, the apparent position of a star to an observer on Earth varies periodically over the course of a year as the Earth's velocity changes as it revolves around the Sun, by a maximum angle of approximately 20 arcseconds in right ascension or declination.
The Copernican heliocentric theory of the Solar System had received confirmation by the observations of Galileo and Tycho Brahe and the mathematical investigations of Kepler and Newton. As early as 1573, Thomas Digges had suggested that parallactic shifting of the stars should occur according to the heliocentric model, and consequently if stellar parallax could be observed it would help confirm this theory. Many observers claimed to have determined such parallaxes, but Tycho Brahe and Giovanni Battista Riccioli concluded that they existed only in the minds of the observers, and were due to instrumental and personal errors. However, in 1680 Jean Picard, in his Voyage d’Uranibourg, stated, as a result of ten years' observations, that Polaris, the Pole Star, exhibited variations in its position amounting to 40″ annually. Some astronomers endeavoured to explain this by parallax, but these attempts failed because the motion differed from that which parallax would produce. John Flamsteed, from measurements made in 1689 and succeeding years with his mural quadrant, similarly concluded that the declination of Polaris was 40″ less in July than in September. Robert Hooke, in 1674, published his observations of γ Draconis, a star of magnitude 2m which passes practically overhead at the latitude of London (hence its observations are largely free from the complex corrections due to atmospheric refraction), and concluded that this star was 23″ more northerly in July than in October.
Consequently, when Bradley and Samuel Molyneux entered this sphere of research in 1725, there was still considerable uncertainty as to whether stellar parallaxes had been observed or not, and it was with the intention of definitely answering this question that they erected a large telescope at Molyneux's house at Kew.3 They decided to reinvestigate the motion of γ Draconis with a telescope constructed by George Graham (1675–1751), a celebrated instrument-maker. This was fixed to a vertical chimney stack in such manner as to permit a small oscillation of the eyepiece, the amount of which (i.e. the deviation from the vertical) was regulated and measured by the introduction of a screw and a plumb line.
The instrument was set up in November 1725, and observations on γ Draconis were made starting in December. The star was observed to move 40″ southwards between September and March, and then reversed its course from March to September.  At the same time, 35 Camelopardalis, a star with a right ascension nearly exactly opposite to that of γ Draconis, was 19" more northerly at the beginning of March than in September. These results were completely unexpected and inexplicable by existing theories.
So as a result of searches for stellar parallax, the aberration of light was discovered by James Bradley.
Bradley continued to research the aberration of light, and made another unexpected discovery, the nutation of the Earth's axis.
Nutation was discovered by James Bradley from a series of observations of stars conducted between 1727 and 1747. These observations were originally intended to demonstrate conclusively the existence of the annual aberration of light, a phenomenon that Bradley had unexpectedly discovered in 1725-6. However, there were some residual discrepancies in the stars' positions that were not explained by aberration, and Bradley suspected that they were caused by nutation taking place over the 18.6 year period of the revolution of the nodes of the Moon's orbit. This was confirmed by his 20-year series of observations, in which he discovered that the celestial pole moved in a slightly flattened ellipse of 18 by 16 arcseconds about its mean position.3
Although Bradley's observations proved the existence of nutation and he intuitively understood that it was caused by the action of the Moon on the rotating Earth, it was left to later mathematicians, d'Alembert and Euler, to develop a more detailed theoretical explanation of the phenomenon.5
As it turned out, the changes in the apparent positions of stars due to the aberration of light and nutation of the Earth's axis are many times larger and easier to detect than even the largest stellar parallax of even the nearest star.
It was not until the 1830s that astronomical instruments became advanced enough that the first stellar parallaxes were detected and measured, after centuries of failed attempts. And the parallaxes of only three stars were measured during the the 1830s.
The large proper motion of Alpha Centauri AB was discovered by Manuel John Johnson, observing from Saint Helena, who informed Thomas Henderson at the Royal Observatory, Cape of Good Hope of it. The parallax of Alpha Centauri was subsequently determined by Henderson from many exacting positional observations of the AB system between April 1832 and May 1833. He withheld his results, however, because he suspected they were too large to be true, but eventually published them in 1839 after Friedrich Wilhelm Bessel released his own accurately determined parallax for 61 Cygni in 1838. For this reason, Alpha Centauri is sometimes considered as the second star to have its distance measured because Henderson's work was not fully acknowledged at first. (The distance of Alpha Centauri from the Earth is now reckoned at 4.396 ly or 41.59 trillion km.)
In 1804 Piazzi reported that 61 Cygni had a very large proper motion and was probably one of the closest stars to Earth, and thus a good candidate for parallax observations. There were many unsuccessful attempts to measure the parallax of 61 Cygni.
When Joseph von Fraunhofer invented a new type of heliometer, Bessel carried out another set of measurements using this device in 1837 and 1838 at Königsberg. He published his findings in 1838 with a value of 369.0±19.1 mas to A and 260.5±18.8 to B, and estimated the center point to be at 313.6±13.6. This corresponds to a distance of about 600,000 astronomical units, or about 10.4 light-years. This was the first direct and reliable measurement of the distance to a star other than the Sun. His measurement was published only shortly before similar parallax measurements of Vega by Friedrich Georg Wilhelm von Struve and Alpha Centauri by Thomas Henderson that same year. Bessel continued to make additional measurements at Königsberg, publishing a total of four complete observational runs, the last in 1868. The best of these placed the center point at 360.2 ±12.1 mas, made during observations in 1849. This is close to the currently accepted value of 287.18 mas (yielding 11.36 light-years).
The distance to Vega can be determined by measuring its parallax shift against the background stars as the Earth orbits the Sun. The first person to publish a star's parallax was Friedrich G. W. von Struve, when he announced a value of 0.125 arcseconds (0.125″) for Vega. Friedrich Bessel was skeptical about Struve's data, and, when Bessel published a parallax of 0.314″ for the star system 61 Cygni, Struve revised his value for Vega's parallax to nearly double the original estimate. This change cast further doubt on Struve's data. Thus most astronomers at the time, including Struve, credited Bessel with the first published parallax result. However, Struve's initial result was actually close to the currently accepted value of 0.129″, as determined by the Hipparcos astrometry satellite.
So the first three measurements of stellar parallax are almost tied for first place.