Rome Vs Sassanids

Rome "proper" had a geographic advantage similar to what the US has today. It was surround by water on 3 sides and mountains on one. That was one the reasons Rome flourished the way it did. Natural barriers made it very hard to invade Rome.

For Parthians to be able to get to Rome proper and be in a shape to fight would have needed a miracle. They only kind of chance they really had was to cause enough damage to Romans army in the plains of Asia to leave Rome completely open. That is why Carrhae could have been turning point in history

Well it depends on how you look at it. The area that is Italy today was surrounded by the sea on three sides and mountains on the other. Mediterranean Sea is not Pacific Ocean but at the same time ancient modes of logistics where nothing comparing to today. This is in sharp contrast to geographical location of Mesopotamian civilizations and later the couples of Persian and Parthian empires for which the center of this empires where located in the middle of cross roads making them open to invasion from one side or another

Europe as a whole is more complex situation. And Yes Parthians would have probably have had a hard time staging real offensive against Rome, that is why I said they only chance they may have had was to cause enough damage when room was on the attack to leave Rome weak and open to attack

Yes, Carrhae was not a turning point, but if the Parthian king had decided to capitalize on the victory and not to turn against his own victorious general it may have turned into one. We will never know

Rome was already in the Middle East they just could not go much beyond the Mediterranean coastline. The logistics to Mesopotamia could not have been much worse than North Africa or England for that matter. The eastern Roman Empire which you used in the example above with Huns and Ottomans had more of presence in the Middle East than Europe

As for china neither Hans nor Parthian/Sassanids had the financial motive or the resources to expand to the east or west into central Asia

The attacks on the empires did just come from other empires. In fact most, middle eastern empires fell to nomads or smaller states surrounding them
Base of ancients and classical economy was agriculture. Empires amassed a large amount of gold by taxing peasants at the same time they were surrounded by nomadic group living in poverty attracted to the gold. They had had to spend a fair amount of cash on the military to defend against such attacks

For Rome this problem was much smaller. The biggest worry may have been tribes of northern and eastern Europe, and because of that they could concentrate on other areas in both military and non-military sense

As for Alexander the Great, I don’t tend to agree with the idea that he was not establishing a new empire because of size. The area he conquered was smaller than the original Persian Empire. He did not need Macedonia troops. Fighting for armies was one of the biggest sources of employment for the ancients, In fact, Both Persian and Macedonians armies had Greek fighters motivated by cash. Alexander had taken enough gold from the Persian Treasury to hire as many troops as he needed.

There are many factors causing Alexander The Great not to be able to maintain the empire. Persians had slowly come to rule the vast area so they learned as they conquered Alexander the Great and his generals were given the empire over night they knew how to fight but not how to rule an empire, but there are probably other reasons

The chief reason for empires to expanded was to increase the tax and resource base, Germany had noting to offer at the time. The attacks on Germanic tribes were mostly defensive on the part of Rome to preempt attack by Germanic tribes in need of money.