I think what people wrongly think about ancient times (and by extension the times of the Islamic Conquests) is that peace was always assumed. Nowadays, we have the UN and a lot of International Conventions that make it so that war has to be declared, but in previous times, war was not declared, peace was declared.
So, if there are two empires and there is no peace treaty between them, they are at war. They will both rob the other's caravans, they will both attack whenever it is beneficial to them, and they will both strive to destroy each other.
So, I want you to imagine yourself in the shoes of Abu Bakr (RA), before he gave the order to invade Persia. Islam was in an active campaign against the Roman Empire already which was impossible to avoid considering that the Arabs relied heavily on trade with Syria, and if Syria was under control of a hostile empire... Rome would hold an uncomfortably large amount of influence over the new Islamic State.
So, there is a war with Rome, a war you know will not stop until the Islam conquers a large amount of territory. This is because, Rome would never accept peace with a small amount of territory conquered by Islam simply out of pride (surrendering to Arab Bedouins is not a good way to get public support). But, once Islam conquers a large amount of territory, they will be forced to respect them as an equal empire, thus the possibility of peace can become a reality.
So, the inevitable result of the war with Rome is:
At this point, both Rome and Islam would come to a rest because of tiredness. But, Islam's geopolitical position is very bad. Now it shares the largest border with Persia, so Persia will undoubtedly attack Islam just like it fought with the Roman Empire for centuries. Furthermore (even with a peace treaty) as soon as Rome rebuilds it will attack Islam to get back Syria and perhaps even to end the whole Islamic threat altogether by taking over Arabia.
So, it was utterly unsustainable to only hold Syria. It was too narrow of an area and surrounded from all sides to effectively defend. So, imagine if you, being in Abu Bakr (RA) shoes, had the opportunity to widen that narrow piece of land to include a much more defensible mountainous and prosperous region, namely Iraq.
And to attack it, you don't need to break any peace treaty because there is none, and it is an empire who is threatening the Arabian Peninsular region as well. And the only reason that empire hasn't already attacked you is that it is currently in the middle of instability which also means that it would be easy to attack.
There is a saying:
It is not bravery but rather stupidity to let your opponent pick up his sword.
Do you want to let the Persian Empire come back to its full strength when you have the choice to stop it from happening?
So, when you live in their world, you have to understand that there is no UN. Peace is not assumed like it is today. There is only one certainty: empires expand.
Also, Abu Bakr (RA) never intended to destroy the Persian Empire. He only wanted to have a defensible position for the Muslims, a position where Arabia is not under constant threat. It was out of geopolitical necessity that he invaded Iraq. And if he had not, Islam would not exist today in any meaningful way.