I have had a different experience with Parthia. I don't bother waiting to build up the economy I just attack simply because I kept going broke even with just a few troops being built. Take Seleucia on the first or second turn and then move the the next Seleucian city. Once there hold the bridge until you get sufficient troop upgrades to take Antioch.

Above Seleucia move your army west to take the rebel city and don't bother making alliance with Armenia because they won't keep it. Move into their territory with what you have and destroy them. When you get to the sea concentrate on trade and replenish your troops. Then move to Pontus. Take their main seaport because it is their main troop production city as well. This will give you Persian Cavalry and another valuable trade route to the rest of the world.

Meanwhile in the far north you should be wiping out scythia. If you move quickly with only horse units you can reach some of their cities before they build walls. If not have infantry bring up the rear. Build lots of watchtowers away from the roads along the way and you can pick your fights. Once you reach the river at Dacia and Thrace put forts on your side of the bridges and it tends to keep the enemy out of you area.

I got to Pontus, destroyed Scythia, and reached Antioch at the same time. For fun I engaged that little hidden barbarian city with the Amazon Chariots. (had never done it before) and I finished off Seleucia from two sides.

EGYPT: Egypt is a paper tiger and i don't care what anyone says. They are easy to destroy because they do not invest in a navy. I build a navy and with Elephants, Mercenary hoplites, and lots of archer/horse archer units I attack each of Egypts coastal cities. I can attack in one turn, destroy the defenders, then move all my troops out of the city and back to the boats except for my general. I then give the city back to Egypt or some faction I'm at war with and then capture the city again. Each time I enslave the population and increase the population of my good cities. Until I can take the pyramids it is going to be too hard to hang on to Egyptian cities. I also destroy all the improvements I can in the city. Then either give the city back to Egypt to prevent them from getting an instant rebellion army or I can give the city to the Roman Senate. I have never seen a city rebel from the Roman Senate EVER nor have I ever seen a faction take a city from the Roman Senate.
Move along the coast and repeat until Egypts major coastal cities have been reduced to 400 each and their ports, and trade, and troop production is devastated. Now you can move in-land and not face as many troops. Take the Pyramids first it makes life a lot easier.

Sigh I wish I could destroy the walls like the Greeks did to Troy.

By now you have taken enough cities that have advance troop buildings in place and you can rebuild your armies and prepare to move west.