As well as releation to the emperor, such as Surena's family and clan were close to that of the Arsacid clan, rulers of the Parthian Empire. However there was a sort of military leadership that went with the Parthians.
The Emperor or Shah e Shah, had usually 18 subordiante kings as well as governors and fedual lords who owed the Great King alliegeiance. They relieded heavily on armored Catarphacts themselves with mounted archers and lightly armored troops, excluding the Greek/Macedonian Mercanaries. Other than that, we're really not sure how the chain of command went for the common troops.