Yes. http://astro.estec.esa.nl/Hipparcos/ has given us probably the most accurate distances to nearby stars to date. The principles on which it worked (the satellite has long since burned up) are a little arcane, but at the core is parallax.
The 'first' methods of determining the distance to the Sun were also parallax, but in an interesting way - a Greek astronomer (Aristarchus?) estimated the distance to the Sun by the shadow of the Earth on the Moon, during a lunar eclipse! (It's simple geometry). However, as several of his input parameters were wrong, he got the distance to the Sun wrong too.
More generally, since the Moon and Sun have the same angular diameter (as evidenced by solar eclipses), some simple geometry shows that once you have one distance (e.g. Earth to Moon), you can get the other. It gets a bit more complicated if you want to get accurate (e.g. orbits aren't perfect circles).
With Newton and accurate orbits of the planets and asteroids, the distance to the Sun was fairly accurately determined; however, only when radar astronomy was able to get an echo (from Eros, IIRC, though it might have been Venus) could the solar system distance scale be nailed down to ~km (or maybe more accurately). Today we have transponders on interplanetary probes, and the 'light return time' gives very accurate distances (we also need good ephemerises). Anyway, the distance to the Sun need not be all accurately known to work out distances to nearby stars ... the parallax angle (pre-Hipparcos) could only be measured to ~1 decimal place!