To a point. Around the Aventine, for example, there has been found a cluster of Palmyrene inscriptions, which indicates that there was a community there. Palmyra was an oasis city in Syria at the center of a number of trade routes that is famous for the strength of its local identity and the visibility of the expression of this identity in other areas. So just because the Palmyrenes were a very visible community does not mean they were unique, and indeed there are scattered literary and epigraphic references to other communities (Tyrean, Ephesian, Narbonensian, etc) resident in Rome.

But these aren't really like Chinatown. Rather these are trade stations. In the pre-modern world, and even today, long distance trade relied on extended networks of interpersonal connection. It would not be a matter of an Ephesian merchant sailing from Ionia to Rome, selling his cargo, and returning. Rather this cargo would follow along channels well established through diaspora communities--perhaps an Ephesian merchant in Rome would make a deal with a Roman merchant, who is willing to deal with Ephesians because his friend dealt with them in the past and found them trustworthy, they would pool their funds together, send a message to an shipper in Ephesus, who would sail to Sicily, pick up a cargo of grain, and go to Rome where it would be purchased and sold by the Ephesian and Roman merchants. These aren't massive immigrant communities like Chinatown, but wherever you see long distance trade you can almost be certain there are these same small, tightly knit and well connected diaspora communities. In aggregate, these made up fairly large and influential segments of the population, contributing to the stunning diversity of the city.

That being said, the closest equivalent was probably the communities of different Italian areas. The different neighborhoods of Rome had very distinct characters, in terms of religious festivals and identity. It is likely that these sometimes to a degree corresponded to different areas of Italy. But outside of that, it is difficult to know whether there was an "Egyptian quarter" or "Spanish quarter". While there were an awful lot of foreigners in the city of Rome, many were likely fairly well integrated into the city itself, not the least because quite a few arrived as slaves.

Good sources for this are Steven Dyson's Rome and taco Terpstra's Trading Communities in the Roman World.