I have heard that there was a time (e.g.: 120 BC) when the Romans had a large influx of slaves, which surely displaced many in the labor market

I for one would really like to know who argues for this (and when this is supposed to have taken place), because to my knowledge no classical scholar ever has. To the best of our understanding slaves made up a significant part of the population of Italy (and a much less significant part of the population of the provinces), but probably not enough to have anything but a social impact. There are reasons for thinking this. First of all, our estimation of the slave population has increasingly been lowered in recent years. Brunt conjectured that as much as 40% of the Italian population by the early Principate might have been enslaved. This figure was already being questioned in the 1980s and 1990s (Harris importantly argued for 17-20% in Roman Italy), but the sources from which we pull demography are few and very hard to interpret. No one, however, has estimated higher, and Scheidel's highly influential 2005 estimate that no more than 20% of the Italian population was enslaved (and significantly less in the provinces) has become more or less the standard. In fact, it is customary to estimate even lower, and Scheidel's revised estimate places the average slave population of the empire at only 6-12%.

As such, simply titanic numbers of slaves would have been necessary to have any real impact on slave labor. The largest single capture of slaves we know of from the Roman world was in 167, when Paullus' army enslaved some 150,000 Epirotes. The interpretation of the Republican census figures is a problem in its own right, split between the High Count and Low Count models (Lo Cascio, de Ligt, and especially Rosenstein are the places to look), and the fluctuations in the census count in the middle of the second century are especially notorious for their demographic impossibility. Nonetheless, by the lowest estimate of the Roman population there would have been at least 300,000 or so adult male and female citizens in Italy in 167. This assumes that the census figures, which are certainly undercounting (by how much we do not know), counted both adult males and females (though it might have counted only adult males, doubling the figures) and assuming that children did not exist, which obviously they did, since they were not of course recorded in the census. This is an enormously unsatisfactory figure that nobody uses, but let's keep it for the sake of argument, since it's far too low to be even remotely possible. In 167 a population equivalent to half of this Roman citizen population that we've just made up was added to the pool of enslaved persons. Many would have been replacing the dead, and the vast majority would have been going to rural Italy--attempts have been made to quantify how many, but we won't worry about it. However, the total population of Italy would have been many times larger, and we're not sure exactly how big it was. The highest estimate of Roman Italy, free and slave, is some 14 million (Lo Cascio, the originator of the "High Count"), and the lowest is Scheidel's recent estimate of 5-6 million. Most estimates hover between about 6 million and 10 million. If Scheidel is right and the population of Italy was very low, the influx of slaves in 167 from Epirus would have amounted to a mere 3% of the population of Italy--demographically significant, but probably not particularly important otherwise. A population of 10 million would have been increased by 1.5% and in the event that Lo Cascio is right and the perceived manpower crisis was actually caused by overpopulation those Epirote slaves would have numbered a mere 1%. This is also assuming that the entire slave population went to Italy, which is certainly not the case. Though the slave population of the provinces was typically lower than Italy's nonetheless the market for enslaved persons encompassed the entire Mediterranean. We rarely have information to determine where slaves were being sold and in what quantities, but certainly not all (or even necessarily most) captives were going to Italy at any given moment.

I am sure there is a parallel to be drawn with modern times and the potential displacement of workers due to automation

There is not, and this is the second problem. All our evidence since Brunt's 1966 study at the very latest indicates that the free and enslaved labor markets did not significantly overlap. Slaves could, theoretically, do anything that a free person could do, but they simply did not. For one thing, slaves were expensive, and a permanent investment. For another, both agricultural work and construction programs, the two major sources of labor where we might see unskilled free and enslaved workers overlapping significantly, were seasonal activities, and therefore had highly fluctuating labor demands. Cato points out the financial problem of keeping excessive slaves: during the harvest season they are employed, but during most of the year when labor demands are much lower, you've got idle (and extremely expensive) mouths to feed. Cato's gruesome financial calculus seems supported, not only by our Roman evidence, but by our evidence from other pre-Green Revolution agrarian societies, which tend to rely much more heavily on readily available free labor than slave labor to fuel the most important parts of the agricultural cycle, which are also of course temporary. Roman Italy was a slave economy, in that there were always slaves to perform certain tasks, but the driving force of the agricultural economy was certainly free laborers. We know, for example, that free labor was being transported enormous distances seasonally. Suetonius tells us a rumor that Vespasian's ancestor made his fortune helping move day-workers^1 every year from Umbria to Sabine country to cultivate the fields, which Suetonius says was in his own time still standard practice. Why pull free labor from so far away when large pools of slaves could be gotten from anywhere? A good summary of our understanding of the Roman unskilled labor market can be found in Ian Harrison's 2008 article "Catiline, Clodius, and Popular Politics at Rome During the 60s and 50s BCE," though I do not believe that his ideas about collegia are still supported. Nonetheless, we seem to have a moving unskilled labor market, which does not appear to have overlapped with the enslaved labor market. A rather touching example of this division in labor markets can be seen in something like the Moretum. In that poem, the poor Italian farmer Symilus does in fact own a single slave, an African woman named Scybale. She is described as custos, "guard," which seems to mean something like a maid here. How realistically a farmer so rustic as Symilus seems to be could have managed to afford and feed a slave is a subject of great debate, but no matter the possible poetic license, Symilus and Scybale do not engage in the same work. Symilus' is the work of the field, whereas Scybale appears mainly to cook and clean while he is out.

This sort of division of labor seems typical, and there seems to be relatively little indication that slaves and unskilled free laborers were competing very often for the same work--they do not typically appear in the same employment in the relatively rare cases when the texts notice them (the extremely inadequate nature of the evidence must be stressed). The strongest evidence we have for a disruption of the labor market by slaves at any time is Plutarch's insistence that Tiberius Gracchus saw in Etruria fields tended solely by groups of foreign slaves, not free laborers. But employment was not Tiberius Gracchus' problem at all, it was the manpower pool of the army, and unskilled, unlanded free workers would not have contributed to that manpower pool in any case. Plutarch's testimony here does not seem to make very much sense on several levels--Rosenstein has a good treatment of the many problems with accepting what Plutarch says about Gracchus' perceived manpower issue. There is certainly no parallel with automation of the modern labor market. For one thing, the modern factory jobs that are the main victims of automation and the ancient labor market have far too many differences to be at all comparable. For another, the whole point of automation is to provide a low-cost alternative to normal labor. Slaves were neither low-cost nor a reasonable alternative to unskilled free labor. Thirdly (though the list of problems hardly ends with only three points), and extremely importantly, the introduction of automation into modern economies is precisely that, an introduction. Roman Italy had, and had always had in its past, a substratum of enslaved workers at all times. Even the largest influxes of slaves were simply adding to a pool of workers that already existed and had an established role in the stable labor market. Even if mass enslavements affected the number of slaves (and there's actually very little evidence from the sources that the Italian economy was ever much effected by mass introductions of prisoners, it seems to be a bit of a non-event in most of our sources) we don't even necessarily have to expect it to disrupt much of anything at all. The number of available slaves can go up as much as we want, but that does not mean that the availability of labor positions for them to fill is moving at all.

1. The Roman world, and the ancient world generally, did not know of the concept of steady salaries and hourly wages, though at least lower administrative staff had the expectation of being paid a rough figure yearly (how this worked exactly we are not sure). As such, all unskilled labor in the Roman world was day-labor.