There you go, give them bows too if possible (think they started out as Armored Horse Archers).
And then you've got an excellent 2nd UU for Persia, besides Immortals, and they thoroughly deserve it, Persia was one of the top 10-15 civilizations in the history of the world in my opinion.
On the other hand, they could include Parthians or Sassanids, they used the cataphract to a much larger extent than Persians did.
Don't you mean the Arsacids and the Sassanids used them more than Achamenids otherwise it sounds like your saying sassanids weren't Persian and that's inaccurate.
Also Zhayedan were Sassanid cavalry the Achamenids did use heavy cavalry but it was a form of chariot archer with the horse and rider clad in scale armor
Then there is infantry the Achamenids had two primary forms of infantry the Sparabara (Shieldbearers) who made up the bulk of the army meant to hold the enemy from going after the archers largely cannon fodder and then there was the Immortals/Companions who served as the personal regiment for the emperor.
The sassanids had Paighans who were essentially cannon fodder and peasant levies alternatively one could have Daylamites who served as heavy infantry as well as Kamandaran archers.
Personaly I think that the setup for sassanians should be:
Leader: Khosrau
UU: Zhayedan chargers ()
UU: Kamandaran archers (Improved composite bowman)
UB: Fire temples (The endorsement of the faith was one of the pillars of the sassanians) Provides additional faith
UA: Deghan reforms (Khosrows attempt at fixing the administration of the empire) +1 gold per district
Leader: Shapur I
UU: Zhayedan archer (Has a melee attack in defense)
UU: Paygan conscripts (Cheaper than the equivalent)
UB: Fire temples (The endorsement of the faith was one of the pillars of the sassanians) Provides additional faith
UA: Captor of Romans: destroyed enemy units have a chance to turn to a builder
Of course we could go with Iranian Intermesso civs or Safavids or even Median (though fairly little is known about them)