The interesting questions of history are sometimes the questions not about what did happen but what did not happen, and why. A strange fact of history is that Buddhism, spreading out of India, spread east but not west. What might have been the reason?
Imagine those monks driving north on the highway out of India through the Kushan Empire. Reaching the Silk Road, they see a signpost pointing right “China” and left “Parthian Empire, Roman Empire”. Why did they only turn left, to China?
Or did they? Maybe, they went both ways but China was somehow a more fertile ground for Buddhism. There are certain similarities between Buddhism in its Mahayana form and Daoism, maybe superficial similarities, but enough maybe to allow Buddhism to get its foot in the door.
Was it suppressed in Parthia? There were several cultures, religions and belief systems there, among them Zoroastrians, Greeks, Jews and Christians. What would have stopped Buddhists from getting a foothold there? The precursors of the Parthian empire, the Persian and Seleucid empires, had always harbored different cultures although Zoroastrianism had dominated.
Was it a philosophical problem? Think of a Mahayana Buddhist monk, with Nagarjuna’s ideas about emptiness in his mind, meeting some Greek guy who was thinking in terms of Aristotle’s ideas. The emptiness of Dharmas, lacking independent reality, versus Aristotelian substances, having it by definition. Was there an incompatibility of philosophies? But there had been a Greek influence in the area that became the Kushan Empire before and this had not stopped Buddhism to gain a strong foothold there.
Or did Buddhism actually spread west, only to be stopped later, when Christianity became dominant in the Roman Empire and later when Islam spread through what had been the Parthian and later the Sassanian Empire? Did Buddhists ever make it beyond Parthia and reach the Roman Empire? If so, why did they not leave a trace, while, on the other side of the continent, having such a strong impact in China? Lots of manuscripts with Buddhist texts where brought from India into China. Why did these texts not reach the Mediterranean or Europe, or, if they ever did, disappear without a trace?