How did the Crucifixion affect Roman-Parthian relations?

The anticipation of the Messiah changed both the Roman and Parthian Empires, the crucifixion more so. Parthia was a confederation of tribes that Roman legions tried to conquer. Parthia was the Super-Power of the East, a deadly rival of Rome's military autocracy.

Parthia trounced mass invasions of Roman legions several times. In 55 BCE Parthian King Orodes slaughtered 40,000 legionnaires under Consul Crassus. In 40 Parthia invaded the Holy Land and deposed the Roman-designated high priest. Parthia soundly defeated Mark Antony in 37 BCE. In 34 BCE Julius Caesar was assassinated mysteriously before his planned invasion of Parthia.

The two empires made peace in 22 BCE under Augustus and Phraates IV (=Arsaces XXII). Why?

Parthia was run by the Arsacid dynasty (250 BCE to 225 CE). Its rulers were composed of Israelite and Jewish exiles (Sons and rulers rosh of ISaac). Secular histories say they were formerly slaves and called themselves Exiles (Parthi in Scythian or Hebrew Galut = Gauls, Galatians, Kelts). They wanted to rebuild the Temple in preparation of the Messiah, dated according to Daniel’s prophecies. (Daniel was member of the Persian Magi, the priestly class of scientists and prophets).

The reconstructed Temple became the most important and biggest construction site of the time (one million man-years work). Vast riches were received from across both empires and around the world (Josephus).
Jesus was identified by genealogy and miracles as the Son of David through Joseph. Mariam (Mary) was of pure Aaronic descent. Jesus was not an itinerant preacher, Protestant style! Pilate acknowledged him as King of the Jews. Pilate recognised Jesus as having a higher rank that he had, as merely a governor of Rome but not a royal. The high priests called Pilate Lord, (Kurios) but Jesus did not.
The resurrection was a shock to both the high-priestly ruling class and the Romans. Ample evidence existed it took place with many Roman witnesses, including Pilate. Based on diplomatic reports of Pilate, Emperor Tiberius acknowledged that Jesus Christ was a god. The Senate, however, refused to ratify this. Tiberius decreed however that anyone who persecuted the Nazarenes (his followers) would be put to death (Tertullian etc).
When Tiberius died in 37 CE (at the hands of Gaius Caligula), Caligula became emperor. He saw that the resurrection undermined the whole Roman/Greek pantheon. Roman gods had to be authorised by the Senate. But here was a Jewish king resurrected and recognised as a god by Tiberius! Caligula, drawn by Egyptian paganism, wanted to be deified as God of this world. In that way he would, he thought, outclass Christ. He planned first to raise a mammoth idol of himself in the Temple of Jerusalem. Why? Because Christ had prophesied that the Abomination of Desolation would stand in the Temple before the end of the Age. This was also prophesied by Daniel (Matt 24:15). Caligula wanted to prove Christ wrong and that he was all-powerful and more god than Jesus. He failed.
The later emperors like Claudius were at first more cautious with the Christian threat, but then introduced a vicious campaign of anti-Semitism, trying to provoke a Jewish revolt. Wars broke out from Britain to Parthia.
James the brother of Jesus was Sagan and Priest of Warfare in the Temple and advised passive resistance. He was killed in the Temple around 62 CE and the high-priestly class then refused to make offerings to the Roman emperor in the Temple. This led to the war with Rome.
it also created a dilemma for Parthia (which was not a military dictatorship like Rome). Suppressing a revolt in Judea did not violate the peace treaty. Titus went further. He destroyed the Temple. He said that that way he would put an end to both Judaism and Christianity (Sacred History of Sulpicius Severus) quoted on NazareneProject site. Titus raised a pagan pillar to his father on or near where the Temple had been.
The Parthian empire lasted nearly two centuries further but on a war-like footing. The Parthian Peace based on the Temple and its Messiah had lasted longer than almost any other in history. The Roman pantheon was destroyed by the resurrection. Rome reverted to Sun worship and foreign cults. The gods like Jupiter, Mars, Minerva became a laughing stock. Rome tried syncretism — trying to combine their paganism with the facts of the resurrection and resurgent Christianity.