In addition to the annual parallax of the star being measured, you also need the diameter of the Earth's orbit, which was also first measured by parallax (together with Kepler's laws, giving the ratio of planets' orbital sizes from their periods):
When Mars came close to Earth in 1672, simultaneous observations by Giovanni Cassini (in Paris) and Jean Richer (in French Guiana), comparing where Mars appeared relative to background stars, yielded a value of an astronomical unit that was about 7% higher than the modern value. [1]
Subsequent measurements of the dual transits of Venus in 1761 and 1769 refined the figure (Lalande, 1771), again using parallax, in a method proposed by Halley. Current measurements are based on radar.
[1] http://curious.astro.cornell.edu/physics/62-our-solar-system...