The flag of Parthia
The original flag of Parthia was simply Parthian blue (light blue, as can be seen on the flag, the color of Parthia) with a purple crown in the middle. This crown perhaps was purple in honor of their former Seleucid masters, seeing as Parthia (and Bactria too) had been vassals of the Seleucid Empire until somewhere around 300 BC, but perhaps it simply was purple because that was and is the color of kings and queens and the likes.
The flag of Seleucid Parthia
After being vassalized by the Seleucid Empire again, a purple border was put on the flag by the Seleucid Empire, because purple was also the color of the Seleucid Empire. But Parthia broke free again, and conquered parts of the Seleucid Empire together with the Greco-Bactrian Kingdom. The border wasn't an ordinary border though, it wasn't on the edge of the flag, but more in the middle (yeah, I can't really explain it, just look at the flags).
The flag of Greco-Bactrian Parthia
They thus used their original flag again, but only for a short time, because the Greco-Bactrian Kingdom then conquered most of Parthia. What remained of Parthia was an useless piece of land belonging to a destroyed nation, and the Greco-Bactrian Kingdom simply vassalized Parthia and changed the Parthian flag again. The purple border was replaced by a red and yellow border, which also wasn't exactly on the edge of the flag.