I think LostInNewDelhi's is the simplest idea.
I think it's also the only practical idea. The alternative to "Sarmatians come to where the Samaritans are" is "Samaritans come to where the Sarmatians are"-- a Khazar-style conversion accomplished by a Samaritan diaspora. But by the time of the Roman-era Israelite diasporas, the Samaritans may be too thin on the ground and the Sarmatians are being assaulted by Huns, Goths, and the coming Turks. Much more feasible to have the Sarmatians invade the Near East at a time when they're strong, and become lords of the Levant-- and like all Levantine lords before them (Egyptian New Kingdom, Assyria, Babylon, Persia, Seleucids) be faced with the problem of Canaan, gateway to Egypt, Arabia, and the Med. Trade must be able to go safely through it, but it's too independent to be ruled by others and too divided to rule itself. At the time of the Seleucid collapse it's divided between an assertive, battle-hardened Judaean monarchy and the scattered Samaritans it's trying to assimilate... so just have the new Sarmatian lords ally with the weaker against the stronger, perhaps turning the area into two or more provinces/puppet states.
There's only one problem, which is probably why this never happened OTL. South of the Caucasus there is a stretch of montane grassland from Erzurum to Tabriz, perfect for pastoralist horsemen. Past that is the ludicrous tax base of Mesopotamia. OTL, Qara Qoyunlu and Aq Qoyunlu were fundamentally mountain/river empires based on this principle. The Safavids kept their capital in or near Iranian Azerbaijan until they moved to Isfahan to be free of the influence of the Qizilbash horsemen of the northwest. All that is well and good... but the Sarmatians are
north of the Caucasus, and few groups seem to have migrated across it. Georgians/Armenians/Azeris stick to their side of the mountains, Circassians/Chechens to theirs. The only examples of groups spanning both sides of the range that I can think of are the Abkhaz (related to Circassians), the Ossetians (who I think had to be forced over the mountains by Circassian pressure?)... and then then there's the trade across Derbent, so Azeris and Jews. But as for powerful hordes? Powerful hordes don't want to spend months/years picking their way through valleys and passes, even if there's great stuff on the other side-- after all, the steppe extends west and east of the Caucasus, so if a stronger neighbor pushes from one direction, just go to the other.
Only desperation and powerlessness would make Sarmatians cross the Caucasus in appreciable numbers, so maybe it's the losing faction of a civil war or something that ends up having to make the trek. Or some knock-ons of the Yuezhi/Kushan migration out of the Xiongnu zone of influence pushes the Sakas west, instead of pushing them south into India like OTL, and the eastern Sarmatians are crushed between them and the western Sarmatians, who fear being attacked by their own dispossessed compatriots (if such a spirit existed among them at all, which it might not have). It may also be a multigenerational process-- one crosses the mountains and takes over Atropatene, the next gets accustomed to diplomacy and efficiently marshaling people/resources, and the third pays the Greeks a visit. The Parthians went through the same process.
To contest what? You're missing a clause there. To contest their dominance? To contest their power?
To contest the end of their sovereignty. Any Sarmatian invasion would be aimed right at their heart (Babylonia) and then try to chase them out of their last refuge (Syria). The Parthians nearly kicked the Seleucids out of Syria too, but were distracted by migrations of the Saka and Yuezhi in the east (who killed one Shah each) and by the time they turned back to Syria, they fell into civil war and Armenia got the prize instead, which they then lost to Rome.