Curious: Persians are much stronger in RoR because of the AE upgrade. This gives their elephants +3 attack and +2 armor (cents are now useless as elephant defense unlike AoE and legions get owned), and, equally as important, siegecraft (so that they take down towers faster than anybody else). In AoE, massed towers + some other unit owned elephants...but in RoR, elephants take out towers so fast, it's sick. Heles and priests take too long to build vs. Persia (same prob with those two units as in AoE...) You can also try the HHA thing, but you'd better have a large enough map...and usually your map won't be large enough. Or, you could just play Carthage, they have 750 hp elephants that can kill Persian eles 1v1 and heles in the midgame which makes it very ugly for Persia. (And Carth actually has a good econ.)
Wedsaz, he is talking about DM here, and the most important resource is the one you didn't mention: time. In terms of power vs. BT, nothing can match a mass of Persian AE early in the game. Support units are used by good Persians as the game progresses. The most important part early is to get a lot of elephants out, and fast. I sometimes get legions for the hell of it with Persia in the midgame because you can pump out a lot of them even with a crap econ, but they die to everything an AE dies to except priests, so they're no use in a rush. (With Persia, you MUST rush.) Cats without ballistics are useless after 5 min or so when hcats start showing up. They're decent support units for Persia if you're going against a hele civ, but that's about it. Priests are a good ele support unit, esp. when going up vs. another ele civ, and something that I think you neglected to mention. The cataphracts and HHA are useless because of the excessive cost for upgrades and lack or range for the HHA, so the fact that they get those units isn't too special.
The problem with Persia, as everyone has mentioned already, is that they're so one-dimensional. With such a crappy econ, they can't possibly hope to keep up in the midgame unless they seriously hurt the opponent in the rush phase. (BTW wedsaz, their econ sucks because you lose if you get into a woodchopping war - and wood is the basis of everything in this game - but more importantly, their non-wheeled villagers are absolute sitting ducks for scythe and archer raids.)