So, something that must be understood before venturing this question is that the relationship between Zoroastrianism and the Vedic tradition underlying Hinduism is inherently antagonistic. This is perhaps a classic example of familiarity breeding contempt - they share cosmology and metaphysics, and agree that righteousness is to be identified as acting in accordance with rta/asha (roughly, the natural order of things, truth). The original point of division, however, seems to have been that where Vedic tradition came to celebrate martial virtues (most clearly embodied in the god Indra), early Zoroastrians seem to have rejected the idea that martial prowess could be inherently virtuous. I go into some of this here.

As I note in the reply to the follow-up question in the linked answer, we actually find mentions in the Vendidad dating to the Parthian period of both several Vedic deities - Indra, Nasatya and Rudra (the last of which would become known as Shiva in Hinduism) - and even Buddha. Except, they're mentioned as daeva (sometimes translated "demons", perhaps better understood as "false deity" or "entity unworthy of worship"). That's just after the Mauryan period. What I'm getting at is, in this period, I really would not expect much in the way of constructive interfaith dialogue. Unfortunately, it's a poorly sourced era in general in that region, and I haven't been able to find any records of interaction. In fact, the 700-page Wiley-Blackwell Companion to Zoroastrianism doesn't even have a heading for Zoroastrianism and Hinduism or anything like that beyond their very earliest shared origins.

So moving on to the second part: it is absolutely certain that the Parsis would have noted the similarities between Avestan and Sanskrit, for the incredibly banal reason that they translated much of their scriptures into Sanskrit (which was later used as a basis for the more well-known Gujarati translations). The two languages are so similar that it would be impossible not to note similarities when doing that.

The aforementioned Wiley-Blackwell has exactly one sentence that really gets to the heart of the question:

For their part, in what is known as the Slokas, in Sanskrit, the Parsis presented their religion as being compatible with Hinduism, though there is a doubt as to whether they were written by a Parsi or a Hindu...

The Slokas are sixteen verses that purport to recall Parsi refugees presenting their religion to the Hindu king Jadi Rana. The WB Companion refers to Schmidt's 1960 "The Sixteen Sanskrit Slokas of Aka Adhyaru", which concludes "On the whole the suspicion cannot be suppressed that the Sixteen Slokas are a spurious product and cannot be considered as authentic" - meaning that they are likely a Hindu forgery. Let's have a look at one of the verses:

2. Who, in the time of bath, meditation, recitation, libation of the fire, eating and excretion, keep the rule of silence mentioned in the scriptures laid down by their own preceptors; who, in this world, venerate the cow with manifold incense, good flowers and plenty of choice fruits: white, brave, heroic, abode of great strength - those are we, the Parsis.

I can definitely see where Schmidt is coming from - the Slokas basically present Zoroastrianism as a sect of Hinduism, mentioning the sanctity of cattle for example (a Zoroastrian would probably be more likely to mention the sanctity of dogs, since cattle-slaughter was not prohibited in Zoroastrianism). That and some philological concerns in the rather mind-numbing text of Schmidt's supports the conclusion that it was written by a Hindu priest with some familiarity with Zoroastrian concepts, in perhaps the 15th to 17th centuries (the earliest manuscript is from the late 17th century.) though dating it is quite controversial.

The other text of note is the Qissah, a Persian epic of sorts from 1599 that narrates the traditional story of the Parsis arriving in Gujarat from Khorasan in the 10th century. The text portrays king Jadi Rana as an extraordinarily tolerant Hindu Raja, who generously received the Parsis who claim to seek asylum from persecution. It features a few sentences where the high priest explains Zoroastrianism to the king that have some superficial similarities with the Slokas - but probably neither is directly derivative of the other (Schmidt goes over some of the arguments back and forth in his rather mind-numbing article). The general tone of the text is a bit different - it portrays the Hindu kingdoms and the Parsis as both threatened by the Muslim invaders, and thus as allies against a common threat. This probably does not actually reflect the experience of the first Parsis who arrived (though the general geographic aspects and so on are generally taken to be correct - but largerly for lack of an alternative narrative), but rather it reflects later tensions during the Muslim invasion of Gujarat, the sackings of Delhi, and so on. (In fact, some other sources like the Rivayat of Adurfarnbag seem to indicate that Muslim rulers were actually more tolerant of certain Parsi practices than Hindu rulers).

Unfortunately the early history and chronology of the Parsis is incredibly foggy. But whatever their initial experiences, these later texts (whatever their purposes) certainly at a minimum indicate a degree of appreciation for their commonalities.

I hope that answered your questions! Feel free to field any followup questions.

Sources and further reading:

Zoroastrians: Their Religious Beliefs and Practices (Boyce)

Wiley-Blackwell Guide to Zoroastrianism (Stausberg et al.)

The Sixteen Sanskrit Slokas of Aka Adhyari (Schmidt)

Qissa-i Sanjan