Parthia is great fun, Horse Archers dominate just about every faction, the only units that they don't absolutely destroy are elephants, and you really don't have to face them that much as the Seleucids die quickly and Romans deal with Carthage for you. You do need to get used to waiting longer to take cities though, you've got no decent infantry so it's best to just siege with a few HAs so the enemy sallies, then slaughter them until you're out of arrows, then return and repeat until they're wiped out.

Opening moves: Move your force from the province north of the caspian sea to the Scythian province right next to it and conquer it, they are generally kept too busy in the west to bother retaking it, eventually they will try but their army is really terrible at attacking cities, just remember to put a couple of Eastern Infantry up there later.

Move your army from your capital/Susa towards Pharapsaa, the Armenians are also trying to take it which leaves their armies in the open. You start with a couple of cataphracts, use them in combination with HA's to run down Armenia's main army outside of Artaxarta, both of their provinces are usually lightly defended at this point, siege them with your horse archers and if they sally, destroy them. Once you take Colchis use its port to build two Biremes to blockade Scythia's ports, build a third once they take Crimea. Have a diplomat near them and get peace as soon as they'll accept it, you'll probably need to beat them at a few sieges in Campus Alanni before they'll agree to it. There is no point in attacking Scythia past that first territory, they are poor, the provinces are vast and sparsely populated, and rebels are everywhere, better to keep them as a buffer against Thrace and the other western powers.

Once the Armenians are taken out, Seleucia is the next best target, take it and Hatra from the Seleucids. Best strategy for this is sending a few HA's to siege, they will sally, you will kill a lot of troops, then retreat. Do this 2-3 times and you will win without having to bother with infantry. Don't bother with Arabia, Egypt will just declare on you for it later, let them have it as it's pretty mediocre. Take all of the Seleucid territory you can, but unless Pontus is attacking you already don't bother with Sardis or Tarsus, Pontus will declare on you to take those every time.

After you've crippled the Seleucids war with Egypt is inevitable. Use horse archers, lots of them. Their chariots may beat your archers 1v1 but you can produce two HA's in the same time they build a chariot. Sidon has high level cavalry stables so you can build Persian Cavalry there, one of the most solid one turn production units in the game imo. Spam as many HA's as you can, siege their cities and lure them into wasting units, keep in mind that if you can maneuver HA's behind phalanx units you will slaughter them. Before pharoah's bowmen the Egyptians have nothing that can stand up to your HA's. Don't bother with the former rebel provinces the Egyptians take, go for Sidon/Jerusalem and then into their African lands. You can come back around to the rebel provinces later, taking them first just wastes your units garrisoning them as they aren't populous or advanced enough to produce units themselves.

From there, you are master of the east and have untold amounts of wealth at your disposal. You're going to win the game at this point so you could do anything you wanted, my favorite options are conquering Anatolia or Africa and heading into Greece or Italy from there.