The main problem for the Sassanids in repelling the Muslim invaders was not that they simply gave up without a fight, but rather that they had just come out of a decades-long war with Rome and were militarily and financially depleted. They were additionally in the midst of a half decade of political instability following the death of Khusro II in 628, with various claimants to the throne fighting among themselves, getting assassinated, and restarting the cycle. It didn't help that Khusro's son, Kavadh II, had ordered all his brothers assassinated after succeeding his father. In short, the Sassanid Empire was at perhaps the weakest it had ever been during Islam's ascent to the world stage. Even so, with all these problems, by the time that the Islamic conquest started in earnest, the Sassanids were still able to put up a more than perfunctory defense.

Yazdegerd III ascended to the throne in 632, around the time that the Muslim invasion of Mesopotamia had started. Initially, the Sassanids dismissed the Arab attacks as little more border raids and relied on local garrisons to fend them off. However, the initial forrays into Mesopotamia in 633 under the competent Khalid ibn al-Walid went well and he took several major Sassanid cities, including Hira, but after he was recalled in 634 and the Arabs suffered a defeat at the Battle of the Bridge, the border remained relatively inactive for some months until the second invasion commenced. The Sassanid showed some astuteness, meanwhile, in arranging an alliance with Heraclius, the emperor of the Romans, and planned simultaneous counter-invasions of Mesopotamia and the Levant, respectively. However, as both empires had plenty of domestic problems to deal with and the fronts were, for the moment, relatively quiet, their attacks were delayed until 636. Seeing his opportunity, Umar moved to attack the Byzantines while the Sassanids were still assembling an army, and defeated them at the Battle of Yarmouk. The Byzantine threat was mitigated.

By the time that Umar turned his attention back to the Sassanids, Yazdegerd III had assembled a large and formidable army, probably the first truly imperial army that had been fielded since the last Byzantine-Sassanid war. With the Byzantines defeated, however, Umar was free to array the majority of his forces against the Sassanids, and later that same year defeated them at the Battle of al-Qadassiyah. With their only imperial army defeated, the Muslims completed their conquest of Mesopotamia in the following years, but even then the Sassanids held out here and there for some time, such as in Ctesiphon, which fell in 637 after a protracted siege.

After the conquest of Mesopotamia, it was really Persian pride and refusal to acquiesce to Arab peace terms that led to their complete conquest in the 640s. However, never did Yazdegerd stop trying to rally his forces and strike back at the Muslims. While the Sassanids never won another decisive victory against the Arabs, it's certainly not fair to say that they simply gave up without a fight.

As for Persian conversion from Zoroastrianism to Islam, in many cases it was on conditions of surrender. The Arabs often offered peaceful resolution of a siege if the defenders adopted Islam, and so this is what many did. Actual large-scale conversion, however, was a gradual process, since non-Muslims were free to keep their faith as long as they paid the jizya tax.