The Parthians we are talking about (fighting the Romans) are what were the earlier Persians and later Sassanids. Their 'overlords' might have Parthian-Scythian ancestors, but they were no longer nomads (sort of like post-Mongol China, I guess).
Not sure where you got your information from, but Persians and Medes were certainly not mainly infantry armies. They probably had a far higher percentage of cavalry in their armies than did the Han, escpecially the Medes.
Medes and Perians did not use chariots (unless you mean the scythed ones used as 'battering ram' by Persians). Assyrians used chariots as did early Chinese or anybody else around that time, then replaced them with cavalry, both bow and spear armed.
Sorry but neither does the Medes or the Persians live off horses and bows like the Huns did, which they grew up with since childhood. The Han cavalry are made up of Huns and nomadic steepe people who have joined the Han side, hence not only have these people grown up living on horses, they are both mercenary and professional in nature.
This is getting off topic, but experiments with light chariots showed that they were very maneuvrable indeed and very stable firing platforms. They also could carry vast amounts of arrows and were not as dependent on flat terrain as we might think. So overall much better than any kind of horse archers in a battle. The reason they dissappeared has more to do with their cost. Heavier chariots were of course less nimble, but they had a different job, just as later cataphract or knight cavalry was not meant to be nimble.
BJ
Only on flat terrain. With a solid axle, a chariot cannot maintain steadiness in fast turns or bumpier terrian. This is not to mention that logically they are expensive to maintain, needed to constantly replace the axles and the wheels.
The Warring States period showed the inferiority of the chariots against pure cavalry. The ancient Chinese armies started with a lot of chariots, which as you know, acts different roles in the battlefield from symbol of leadership, to observation platform, to fire support platform. As some of the Chinese states started to engage the Hun, the concept of pure cavalry army began to be implemented, proving outright superior to the chariots. One of the advantages of the winning Qin army, is that they came from the west side, and so they had many horse riders with them.