This is tricky. The most obvious way is to get Rome out of the picture. The problem with that is, Roman intervention in the east is arguably what weakened the Seleucid Empire enough in the first place to where the Parthians could roll them over. Without the Romans, the Seleucid Empire will be much stronger (think of what Antiochus III could do without Roman interference), and likely able to prevent any significant Parthian gains at the very least, and possibly reverse them in time. But then the Romans, at that point, are unlikely to not expand into the east themselves, and pound for pound, they will almost always be able to check any Parthia expansion further west.
So the solution is to get Rome to collapse, or at least severely debilitate them, between the 180s BCE and 60 BCE. The most obvious way to do that is around the mid-late Republic, during and slightly after the Cimbrian Wars. You had a lot of things happening either simultaneously or in quick succession; the Cimbrian War, the Social War, the Roman civil war, the Mithradatic War, the Spartacus revolt, more civil war (Sertorius in Spain). You could make a lethal cocktail of any of those (the Cimbrian War is a particularly potent POD), and shatter Roman hegemony in the east, if not actually destroy the Roman state. From there, it wouldn't be too implausibe for the Parthians to eventually fill in the vacuum, though they'd probably have to deal with strong Pontic and Armenian states. These could be overcome though, in a way the Romans probably could not.
Beyond that, you have to have Parthia develop a more centralized political system, otherwise this empire is going to be even more unstable than the OTL Parthian Empire was.