So I can talk about this from the Chinese side in the respect of the Later Han and the three kingdoms civil war that followed (so 25-280 CE) era. I and u/10thousand_stars have set out Rome (known as Daqin)-China contact for this period (and 10thousand_stars for the Jin dynasty after that). Hopefully, someone can provide a Roman and Parthian (known as Anxi) perspective or from other periods China.

I think some dealing with the perception of Parthia's actions towards China as a blocker might be skewing the rest of the question? As linked answer shows, there were meetings between Romans and China's court though only two Chinese envoy trips were attempted and while both were recorded to have failed, one was quite simply the Wu envoy Liu Xian died on the long journey there. The other was Gan Ying in 97 where he was stopped at Tiaozhi (exact location disputed) on the Persian Gulf as told it would take two years by the sea. While that was involving a rather sizeable omission of the overland trade routes as a possibility, that was probably local figures blocking a potential loss of trade rather than a cunning plan at the Parthian court.

During the over two centuries, that is simply one block by anyone that might be deemed Parthia, one made at a local level with lie about travels routes rather than any grand policy to curb miliatry threat. China and Rome's bigger problem (then Parthia being in the way of one route of contact) with getting into contact was the rather large distances between them and from the Chinese side, there wasn't any urgency to do this. The Later Han tried once and that was on Protector General of the Western Regions Ban Chao's own initiative and a return envoy by Wu Emperor Sun Quan over a century later. Rome's rare envoys were probably not actually envoys from Roman Emperors but merchants seeking to expand their trade and for which the Chinese courts dressed up as being paid tribute to by a distant empire to add to their prestige. Sending an envoy a such a long distance to a very distant power when they had troubles at home, including deteriorating position on it's northern borders, was not a priority for Han China or for the warlord states that followed it's collapse.

In terms of Chinese-Parthian relations during this time ie the main question, interactions were brief, there was no fighting or splitting of lands. Later Han had expanded into the Western Regions, including Tarim Basin, under Ban Chao from 78 CE onwards and that drew envoys from Parthia, probably merchants like the Romans rather than sent from Patharian court, in 89 (lions and antelopes were given to the court) and 101 (lions and ostriches). These were welcome for the court with distant "tribute" adding the glory and legitimacy of the regime, but Ban Chao retired in 102 and Later Han's fragile position in the region was gone by 107 after major revolts further compounded by events in Liang, Ban Yong's restoration of a position in the 120's only partial. With the retreat in 107, the envoys (such as they were) stopped.

Even during the Han's brief time in the Western regions, what contact there was seems to have been merchants arriving at Han court and dealings at a local level of one staff officer sent by his commander. Not a established relationship between two courts with foreign policy alliances against those in the Tarmin Basin, or miliatry clashes and it would have taken time for the latter to be on the cards.

Now I can't speak of how the Parthia court felt and if they misread the Han's strength during Ban Chao's flourish but the Han, even in the unlikely circumstances they entered an alliance with such a distant power as Rome, was not in a position to be a threat for awhile. Even before it's collapse into civil war in 190 CE.

First the Han (or northern successor the Wei dynasty) would need to have gone back to stretching increasingly declining resources towards the Western Regions again, despite the finical pressures and the decline in the north. Doing that, then the merchants might have returned as "envoys". To be in a position to engage in a war or reach Parthian lands, The Kushans (Great Yuezhi) in Afghanistan and Pakistan and other local powers would need to be dealt with. Let alone nearer threats like the Qiang whose revolt had forced the withdrawal from the Western Region. There also doesn't seem to have been any inclination towards devoting major resources to such a distant fight as even with Ban Chao's expansions in the Western Region, he was operating with minimal support, often acting on his own initiative while relying on local troops.

So in summary, during this time, Han China and Rome did have contact via merchants, as did Han and Parthia. There was one recorded attempt to block Gan Ying by local Parthians but that was over trade. The Han doesn't seem to have been in a position to be a threat to Parthia or to be a practical miliatry ally to Rome against them. What relations between Later Han and Parthia were on a merchant to Han court or a local level, there was trade but Parthia wasn't on the minds for miliatry expansion or Han foreign policy.

I hope help this helped and you have a good weekend

Sources

Fire Over Luoyang by Rafe De Crespigny

Northern Frontier by Rafe De Crespigny

The Roman Empire According to Chinese Sources by Kristina Hoppal

Ruler of the Treasure Country : the Image of the Roman Empire in Chinese Society from the First to the Fourth Century AD by Lin Ying