Precision in the measurement of the distance to the Sun How precisely can the distance to the Sun be measured? Wikipedia says the distance to the Moon can be measured upto millimeter precision. But Wikipedia article on distance to the Sun says only about Astronomical Unit and nothing on the precision of the measurement of the distance to the Sun. I am certainly aware that we now use radar to measure the distance to the Sun, and I remember reading the precision in its measurement somewhere on the Internet, but I can't find it anymore. When I try to find the relevant information on the Internet, all I find is educational articles on things such as parallax, which is certainly superceded by radar measurements, or articles on lunar distance, which I am not looking for. Relevant references will highly be appreciated.

Edit: I just found out on Internet that we measure the distance to the Sun through distance to the Venus or Mercury. Anyway I want to know the precisions or errors in these measurements..
This is a partial answer too long for a comment, but it may help to get the ball rolling...
Starting from this answer we can see that the standard gravitational parameter of the Sun used by JLP in their development ephemerides is 1.32712440040944E+20. While that doesn't mean we know the Earth's orbit to 1 part in $10^{14}$ it hints that it's believed to be known surprisingly well!
It's possible that something like this might shed some light on the subject, but it's not an easy question to answer. The Planetary and Lunar Ephemeris DE 430 and 431, IPN Progress Report 42-196 (February 2014)  Part of the problem is that we don't know exactly where the solar system barycenter is with respect to the Sun because there may be bodies far away we haven't detected yet. That's only a small effect on the Sun-Earth orbit. If I had to guess, I'd say it's somewhere between 1 and 100 meters uncertainty between the Earth and the center of the Sun. Where the edge of the Sun falls is a horse of a different color!