IMO he came very close to destroying the Parthian Empire. I myself don't really know if he'd be able to invade India, but bring down Parthia, possibly. But how far into did he go? Was it only the areas of Mesopatamia? Or did he enter Iran proper?

Highly unlikely. It took even Alexander a lot to conquer a western fragment of India. I also think such a campaign would be useless anyway. Rome wouldn't benefit from such Imperial Overstretch. But aside from that, it's just impossible, given all the reasons mentioned above.

Oh I wasn't talking about invading the WHOLE of India, I meant like a little piece of Pakistan (modern Pakistan) that Alexander conquered. But I truly think he would have been able of destroying the entire Parthian civilization, that I am sure of. Although he may have not been able to hold all of Parthia.

The capture of Ctesiphon marked the official culmination of Trajan's third campaign and the war.   When he arrived in Charax (now modern Basra), there is the story that he'd arrived just in time to see a merchant ship sailing for India and he publically lamented he was too old to follow in Alexander the Great's footsteps. 

If true, this would indicate even he didn't think it was possible.

Trajan was at least 60 and suffered a stroke while in Persia.  He died on route back to Rome.

He would have had to have started far, far earlier in his career if he stood a chance at India

As YusakuJon3 mentioned, traders from the Roman Empire were well known in India.  The Indians referred to them as "Yavana", which was the name they originally gave to the Greeks who ruled parts of the northwest Indian subontinent in the centuries immediately following Alexander's invasion.  "Yavana" is believed to correspond to "Ionian."

It's interesting to note that some of the "Yavana" Romans of the Indian port cities had a reputation as skilled fighters and were sought by local Indian rulers to be hired as mercenaries. 

There does not appear to be any evidence, however, that any Roman Emperor undertook any military expedition that reached India.

It wasn't a failure, he successfully took Mesopatamia. Parthia did not take back the region until Hadrian himself ABANDONED the region completely. Besides, Hatra was not a defeat, it was just a fort that did not surrender to the Romans at all. He had to halt his invasion because of the Judaean revolt in Palestine, and because the Mesopatamians also revolted. It wasn't a defeat just because Trajan's health failed him! He conquered additional territory, yeah he did not take over all of Parthia but Alexander III wouldn't have been able to take over all of the Persian Empire if he had health problems too.

Deriving from Persian "Yauna" which of course derives from the Greek "Ion".

Interestingly all western nations know us Hellenes as Greeks (Greece, Grecia, Griechenland etc) because the Romans prevailed while eastern nations know us as Iones (Yunanistan, Yunan etc...) because of the Persians.

As fascinating as it would have been, I don't think it would have been possible.

"It wasn't a failure, he successfully took Mesopatamia."

The goal of Trajan was not just to conquer Mesopatamia, but the rest of Iran as well, and in that aspect he failed, whether due to health or defeat.

"Oh I wasn't talking about invading the WHOLE of India, I meant like a little piece of Pakistan (modern Pakistan) that Alexander conquered. But I truly think he would have been able of destroying the entire Parthian civilization, that I am sure of. Although he may have not been able to hold all of Parthia."

Not even possible, north western India at this time is held by the Kushans a greater power than Parthia, and perhaps under the great king Kanishaka who subjugated most of western central asia and has hundreds of thousand of troops whose quality is no way inferior to the Legions.