Just to add a few points:

Macedonia has one of the very best cavalry and some nice archers.
The Seleucids have a very strong cavalry, including elephants and kathaphracts. They also own normal archers.
Parthia is very good in cavalry of all shapes. Too bad their infantry is a mess.

If you like great cavalry and good archers, you should play BI expansion pack. Related to the fall of the roman empire by moving hordes of mostly mounted barbarian tribes, you will find outstanding cavalry here.
All the nomad steppe fractions have light mounted archers which can do a good rain of arrows against anybody and are nice in pursuiting enemy units. But don't try to attack when the enemy is not weaken. You may compare them with Scythian mercenaries of vanilla RTW.
Any steppe fraction also has lancers, in parts similar to macedonian lancers but with better performance. They are fast and do a lot of damage in their first contact. If you don't break through with that it is better to pull them back and make another full scale assault.
Noble cavalry is always good as a shock-cavalry and there are many.
Also a great bonus for any eastern fraction including Sassanids and Eastern Roman Empire are foot archers with composite bows. They fire over longer distances than any western archer and perform outstanding!
The Sassanids are the heirs of the Parthians and can use most of their best cavalry units. Their heavy cavalry is maybe the best regular cavalry you can find in this game. While good armoured, equipped with maces, they can also use strong bows to fire missles. They are nothing less but outstanding!