I've answered a similar query here.

The Achaemenids and Sasanians both extended a fair bit into the Indian subcontinent and certainly exterted indirect influence on the Gangetic Plain societies of their day, even though their territories only reached about through modern-day Pakistan. Beyond that, in the Achaemenid era, remember that the Achaemenids largely depended on pre-existing dynasties to replace with their own in most of their conquests; in that age, there wasn't really a precedent for unifying India, that only came toward the end of the Achaemenid era. Geographically, projecting power beyond the Indus River was a very tall proposition in any event.

For the Parthians, they never seem to have had an effective enough state apparatus for that kind of conquest beyond their early expansions; they lost their Indian territories to the Kushan.

For the Sasanians, they faced a formidable opponent in the form of the Gupta Empire, with whom they certainly seem to have clashed.

There were powerful enemies in the way and they were at the end of a long and difficult logistics chain from their core territories. The Indus valley seems to have generally been the limit for armies whose main heartland was Persia/Iran. In many cases it wasn't united and could be defeated piecemeal, and it wasn't too far to manage logistically (particularly if you could control the coastline and bring supplies in by sea and then up the rivers. That last isn't so feasible with the Ganges, and the states along the Ganges were typically larger and presented tougher resistance.

The people who had more success including conquering parts of the Ganges area were usually from central Asia. The ones closest in time to the OP's question would be the Bactrian Greeks, who expanded from their original kingdom(s) in what's now Afghanistan and Kyrgyzstan into the Indus valley and the upper reached of the Ganges as the Maurya declined; the Saka nomads who established a kingdom in the southern part of the Indus region; and the Kushan, who had pushed the Greco-Bactrians and Saka out of their territories in Seistan and Bactria and then moved into the Indus themselves - and fought wars with the PArthians and Sassanids, though they were eventually pushed out of India proper by the Gupta empire (and then lost their central Asian territories to the Gok Turks.