What if the Persian Empire manages to attack the heart of the Roman Empire at some weak point in Roman history and besieges the eternal City itself ?
That's a good question. I mean you speak about "attack" and "besiege".
You do not mean "occupy" or "take".
I think that it is possible only with alternate Persian Empire. OTL Persian Empire is out of the question.
The good PoD for alternate Persian Empire would be the death of Alexander the Great during battle of the Granicus.
No, it'd better be
Battle of Issus, the Persians needed to be properly scared in order to pay better attention to their Western border in the future.
So, Alexander is killed by a Persian arrow, the Macedonians and Greeks are cutting each other throats. The Persians are triumphant, restore their Empire and since then they are aware that a European army might invade Asia with disastrous consequences.
So this alternate Persian Empire has it's ups and downs. And it
might happen that during the italian campaign of Hannibal the Persian Empire would be as it often was - the world power.
So this Persian Empire sees once again a highly aggressive European power which is an obvious threat to them. And after the battle of Cannes they might decide to send a Grand Fleet with an army of 20-30 000 into Italy.
Why not?
The point would not be to occupy, to conquer or take.
The point might be to besiege Rome without any intention to actually take it.
The intention of this Persian campaign in Italy might be to force the Romans to come to terms with Carthage and Hannibal. So that Rome would not be eliminated as an independent power; it might be greatly reduced. The Persians would not be interested in seeing the only great power in the West - Carthage or Rome. They would prefer two powers at least in order to play them against each other.