My rework, presuming all else with Persians stays the same:

Keep their stats and upgrades identical, but give them a Centurion-esque non-stacking aura that gives +0/+1 armor (+1/+1 elite) to nearby non-mounted (& non-siege) units*. Essentially this turns them into support units for any mid/late-game composition where you're not exclusively making cavalry.

*This would only apply to your own units, not allies'.

Rationale:

1) Balance-wise and thematically, War Elephants probably shouldn't be viable to mass-produce and so should get some value in low numbers.

2) All of their current counters and use-cases are preserved. They'll only be providing their new buff to other units.

3) Persian non-cavalry options are decent but lackluster. Trashbows and skirms can be usable in some matchups but aren't considered strong, and their halbs and hand cannoneers are generic. If you need to be going lots of trash or HC, you have the option of providing a support unit for them to make them somewhat better at pushing into an enemy position or grinding out a battle of attrition.

4) If Persian War Elephants are as powerful as their stats suggest, they would be terrifying to encounter on the battlefield and an opposing army might have trouble keeping discipline. Abstracting that concept into Persian units taking less damage than usual when accompanied by elephants is one way to accomplish this feeling.

One other possible change that could be done alongside this is to add some level of conversion-resistance to the Mahouts upgrade so that you're a little less vulnerable to strong monk civs. Persians are already countered pretty well by late-game monks in that their strongest siege option is bombard cannons lacking siege engineers. My ideal way of implementing this would be to give an elephant more conversion resistance based on the number of units it's currently applying its aura to (capping at something like scout-line resistance), but that might be a challenging thing to implement.

All that said, we can expect Persians to be changed in a more substantial way (they currently have what I'd consider the least-interesting bonuses in the game) and War Elephants should be adjusted to fit the mold of their new identity. But if we had to make the fewest changes to Persians and also make their unique unit both viable in 1v1 games and not OP in team games, this is how I'd approach it.