Well...Copernicus was able to use his heliocentric model to establish the relative distances of the planets compared to the Earth-Sun distance, and he got the answers for the planets Mercury - Saturn to within a few percent back in the early 1500's when De Revolutionibus was finally published in 1534. Now, all you had to do was use the parallax technique to determine from an angular shift, the distance to one of the bodies in the solar system. But, a simple calculation shows that the parallactic shift using a terrestrial distance as a baseline, is almost impossible to measure with the tools available in the 16th and 17th centuries. In principle, you would have to use simultaneous observations at two points on the Earth, on opposite sides of the Earth, to see even a modest shift in the location of Venus or Mars relative to the fixed background stars.
A number of estimates for it were obtained by Aristarchus and Ptolemy of around 4.8 million miles from geometric considerations. Kepler estimated 14 million miles, but it wasn't until Cassini in the mid 1600's that the first sensible estimate of 86 million miles was established using careful parallax observations from two locations on the Earth. Then in the 1700's, Encke used the transit of the planet Mercury to get 95,279,000 miles. This was a very direct method because the orbit of Mercury was well known, and timing the ingress and egress of the eclipse was a simple matter.
The first high-precision measurements were not made until 1900-1901 when an international team of astronomers measured the distance to the newly- discovered asteroid Eros which at that time was 14 million miles from the Earth. By using the Earth's diameter as a baseline, and knowing the orbital parameters of Eros relative to the Earth-Sun distance, they were able to establish a precise measurement for the Earth-Sun distance. Today, this is done by bouncing radar pulses off of Venus and timing the signal return delay.
Over 2842 observations of the parallax of Eros were made from 30 different telescopes at 24 different observatories to get the solar parallax of 9.790 +/- 0.001 arcseconds. With the known Earth radius, it was then a simple matter to get 1 AU = 149,675,000 +/1 17,000 kilometers or 93,004,000 +/- 11,000 miles.