Why does Astronomy still use the Anno Domini system for Time Synchronization
What makes you think that it is "obviously not 2013 on Earth"?

In actual calculations, astronomers use the Julian day, which is a decimal representation of time. A Julian year is exactly 365.25 days of 86,400 seconds each. Astronomical coordinates are usually written in the J2000 epoch, which allows us to compensate for Earth's axial precession.

Our ability to measure and record time is limited only by the accuracy of our most precise clocks. There's nothing fundamentally wrong or imprecise about it.