So, let's say Jesus' story somehow or other plays out about the same way, and Christianity begins in Parthia.
But I'm not sure Christianity would rise to the forefront among the Parthians. It has to compete with Judaism, Zoroastrianism, and all sorts of Hellenic and Iranian gods, but more importantly, I don't think martyrdom would be as widespread among the Parthians. OTL, martyrdom became an important theme in Christianity, and public martyrdom may have acted as a catalyst for conversion because martyrs' fearlessness impressed people. But the Parthians just weren't as big on huge public displays as the Romans, so the Christians would get much less publicity. It'd probably, as you said, just be considered a weird Jewish or Zoroastrian sect.
The Romans were very accepting of any religion that wasn't a challenge to imperial authority (monotheistic Christians and Jews, rebellious druids), a public menace (druids for human sacrifice and Bacchus' cult for being Bacchus' cult). If Christianity somehow rises to the forefront in the East, the Romans come to see Christianity as a Parthian/Sassanid religion, it may be considered the religion of the enemy and never gain traction in the Empire. Christianity would, in the end, be a predominately Mideastern religion and not a predominately European one. It would also have more trouble surviving in closer contact with... well, if you ignore butterflies, Islam and Mongols, but it'd probably face similar challenges. And Europe, being pagan, may have more trouble advancing from the Dark Ages if and when Rome collapses without the unifying force of Christianity.