Plenty. Restricting ‘ancient’ to the pre-Islamic era (or listing examples of Indo-Persian interactions would take forever), the first Persian (Achaemenid) Empire literally conquered parts of what is now Punjab - this included the ancient Indian academic centre of Taxila - and our word ‘India’ itself comes from Persian via Greek, with the oldest known attestation being from Darius’ stele listing the satrapies of his empire: Sanskrit /s/ (as in ‘Sindh’) corresponds to Iranian /h/ (as in ‘Hind’, land of the ‘Indus’, or ‘India’, rendered ‘Ind-‘ in Greek).

It is not universally but very largely accepted that it was the Achaemenid’s use of written Imperial Aramaic that spread writing to India again (over a millennium after the Indus Valley script died out), with the Kharoshthi and Brahmi scripts based on that - the latter being the ancestor of all major Indic writing systems today.

Multiple other empires encompassed both Persian and Indian territory: Alexander the Great defeated Porus in what is now the Punjab, Kanishka’s Kushan empire, the so-called ‘Alchon Huns’ conquered parts of both the Sassanid and Gupta Empires. Other peoples coming in from Central Asia fought with both the Persians and Indians.

Buddhism - of course an Indian religion - spread to the Parthian Empire and was large enough a minority religion to see special edicts repressing it under the Sassanids.

There was of course a great deal of trade too: we can thank Indian trade with Persia for the appearance of several spices, as well as citrus fruits, and chickens and the aubergine (eggplant) in the West and most of the wider world: e.g., citrus and chickens both reached India from SE Asia in prehistoric times, and later Persia - chicken was known as the ‘Persian bird’ by the Greeks, and the English words ‘orange’ and ‘aubergine’ both descend from Dravidian words via Sanskrit and then Persian. Slightly more controversially, many linguists believe the word ‘rice’ may have taken a very similar route.

This is not to mention the indirect association both Sanskrit (and its descendants) and Iranian languages have of being quite close Indo-Iranian sister languages, a subgroup of Indo-European with closely related languages with a common ancestor in Central Asia significantly more recent than Indo-European, with influences also on religion (comparing many Vedic elements of Hinduism and Zoroastrianism - in fact we can regard ‘Vedism’ and Zoroastrianism as sister religions) and in large part corresponding archaeologically to the Andronovo culture and Bactria-Margiana Archaeological Complex.

I’m aware some of the points raised here, largely the consensus or close to it among Western historians, are not universally accepted among some Indian historians, and feeling lazy right now… but will provide sources when I can if asked. Hope this gave some introduction.