The first thing to keep in mind was that Columbus plan to go east by going west was completely bonkers. The intellectual world of the fifteenth century had a very good idea of how large the world was, and thus they had an accurate idea of how far away China was from the Canaries, the staging ground for Columbus--about 20,000 km. Columbus, due to a series of miscalculations including a misunderstanding of how long an Arabic mile was, believed it to be 3,000 km away. So the first reason why nobody did what Columbus did was because Columbus was not a brilliant vision nary navigator, but a very, very confused person who had no idea what he was getting into.

The second factor is that the "Parthian price" is something of a myth that has largely been rejected by modern scholars (eg Gary K Young's Rome's Eastern Trade). It is something I have seen in Chinese sources but not really Roman ones. One thing to bear in mind is that the primary trade connection between Rome and India and China was not overland, but rather by sea--even today seaborne transport is significant cheaper than overland, and I have seen estimates of ancient land transport being twenty seven times more expensive than maritime. This primarily proceeded out of Egypt's Red Sea ports and took a direct route to the Indian coast, which through careful timing allowed a round trip to take under a year. Landborn transport was not nothing but it was secondary.

The other reason is that the view of the Romans and Parthians being locked in a constant struggle is not really tenable. There were a few wars, but these were sporadic and focused on the very narrow political questions of Armenia rather than a titanic "cold war" between superpowers. And after Nero and Vologases hashed out an agreement over Armenia the two were at peace until Trajan decided he needed another notch on his belt. The accord between them is best seen in the city of Palmyra, which moved fluidly between the two without any sense of conflict.

I think the best popular book on Rome's eastern Trade is Raoul McLaughlin's Rome and the Distant East. It contains plenty one could quibble with, but manages to synthesize a huge amount of information while remaining very readable.

Hi, this is a great question, and I'd love to make this my first official answer on this board (I've lurked on reddit and r/askhistorians for ages though).

I think there are several key factors to this:

The first is that the Romans were never truly as interested in going as far as China as we might expect an expansionist power like Rome to be. Especially in the Imperial period, which is when the evidence we have of Roman knowledge of China and the far east dates from, the Imperial Roman world, and economy in particular, was essentially self-contained. Its one of the factors contributing to the much slower rate of expansion in the principate and dominate than the Republic. The economics of the Empire was entirely based in dominating lands which the empire already controlled and in which they were better able to control; further there existed a circular system of financial reciprocity whereby the noble families would be granted lands, financial gifts, etc by the Imperial household which they would then "return" as a "gift" to the imperial household or emperor himself as part of their wills on their death, and so on and so forth ad infinitum Wheeler (1993), Dmitriev (2009) and Sidebottom (2005) all discuss this and the Roman expansionist trajectory and its financial underpinnings quite well.

Second, and related directly to the above point, Rome already had well-defined trade routes and trade relationships which played a significant part in determining where and when Rome expanded, with whom she allied, etc. For example, Roman expansion into Nabataea, Mesopotamia, and the Syrian desert were specifically to gain greater control over the transit and trade routes from Parthia to the ports in the Mediterranean (and thus, the Roman heartland). Its one of the reasons why Rome expanded so much effort in maintaining control over Syria and Judaea at the time, because their most important trade routes passed through these regions (especially southern syria and northern judaea, and the area of the dekapolis) and not exerting control, either formally or informally, meant that they would have to rely on other means to secure the trade they wanted/needed.

Which leads to point three: Despite controlling the entire Mediterranean basin, as well as coastlines on the Atlantic parts of Europe past the pillars of Hercules, and the black sea, bosporus, and the red sea, for most of its history, Romans were notoriously terrible sailors. There's the tale Polybius tells us in book 1 of Histories that in order to defeat the Carthaginian navy they had to capture Carthaginian naval technology and reverse-engineer it backwards, and learn quite hastily how to use it effectively. That's not to say all Roman citizens were terrible sailors, after all teh Greeks and Phoenicians were absorbed into the empire and their knowledge was not-too-slowly assimilated into the Roman arsenal and allowed Romans to have great trade fleets crossing the Mediterranean; but there is a reason why Roman generals always preferred land warfare to engagement on the sea. There are also stories of various groups of pirates thoughout the Roman period based in Cilicia and other traditionally sea-faring locales which the Romans had great difficulty in putting a stop too, because they were inferior sailors. Then there's also the fact that Romans were some of the first people to develop fish-farming. Partially, this was surely because of simple demand: the Romans LOVED seafood and if you could farm it, it becomes a more stable "crop" to sell at market; but surely part of this is also due to the fact that they were simply inferior sailors and it was easier for them to farm the fish rather than to catch them at sea.

We do have some evidence of long-distance Roman trade, based out of the ports of Arinoe, Berenice, and Myos Hormos in Egypt (and potentially another "emporium" city in Aksum) focused almost entirely on the spice trade, in which they would make stops in Yemen, some Roman-allied cities bordering Parthia, and then eventually India (in particular Barigaza), according to the Periplus of the Erythraean Sea. But this is very little compared to the sea trade and exploration of teh classical Greeks and Phoenicians, or of the "trade fleets" the Romans had criss-crossing the Mediterraean or the major river systems of Roman Europe. Between this overland trade via the Syro-Mesopotamian deserts, and the small amount of sea trade between Egypt and India, they most likely either didn't want or need any more direct contact with China (or other east asian peoples).

Further, there seems to have been some, albeit fleeting and minor, contact between Rome and the Han Empire. Several Geographic writers of the time including Strabo and Claudius Ptolemy are aware of its existence (though not so much about major details), and likewise Chinese authors are aware of the existence of "weilue" or a great empire in the west. They of course would have had contact with the Graeco-Roman world view the Hellenistic Seleukid, Bactrian, Sogdian, and Indo-Greek kingdoms, many of whom would have had trade relations, if not direct diplomatic embassies with each other at various times (particularly during the Qin dynasty). There may have been some direct travel as well, with Florus recording embassies of several groups, including those potentially identifiable to the Chinese, to the Emperor Augustus, though there are no other accounts of such a mission and if they actually existed they were probably private traders rather than diplomats. The Periplus of the Erythraean Sea notes that the people of "Thinae" (Sinae; China) were rarely seen (in the mediterranean-based world of the writer) due to the fact that at the time of its writing (1st C) it was extremely difficult to reach its borders (even considering the relative ease of getting to other places in Asia, such as India, by that time). The only account of Roman travel to the Chinese far east is that of one single well-travelled man, Maes Titianus (who is himself described as "Macedonian" by Claudius Ptolemy, but whom was actually probably of Semitic origin, since his name Maeš is of Semitic origin) who reached, at his farthest extent, the "stone tower" which has been identified by several scholars (including Cary, Hill, and Stein) as being either the "Tashkurgan" of the Pamirs, or western Xinjiang.

So, in conclusion, I think the best reason to answer your question is mostly because they didn't really have a need to "pull a columbus". They had their trade with the far east covered through controlling the trade routes overland coming via parthia, as well as (a small %) coming from sea trade via India.