The last few years of Trajan's life were consumed by a massive and highly succesful (on the battlefield, at least) war against Parthia--in a scant two years he drove down through Mesopotamia and sacked the capital of Ctesiphon, in one of the great achievements of Roman arms, although very poorly understood due to the paucity of sources. The conquests did not stick because the whole thing was an ill conceived fantasy by an old warmonger,^1 but at one point his armies did reach the Gulf, at which point, according to the historian Cassius Dio, he saw a ship departing for India, and said wistfully that he wished he were young so that he could follow in the footsteps of Alexander the Great.

That is probably the source of the claim, which is quite false. The Romans already had a direct trade route to India, through their Egyptian Red Sea ports, and these conducted a fairly brisk level of trade for the entire first century CE. Trade also came through the Persian Gulf, but it was complementary to rather than in competition with the Red Sea, as the due to the nature of sailing patterns the two routes reached the Mediterranean at different times of the year. There is no evidence that the Parthian War conducted by Trajan was part of a trading conflict, if for no other reason than that merchants got along quite well before the conflict erupted.

^1 Quick note that I am being a little facetious here. It is hard to know what his motivations were in beginning the war, but equally hard to ignore that Roman Parthian relations had been just fine for over half a century before.