The original Persians (initially the mobeds, women and kids, and slowly the men) who came to India from 700 AD to 1000 AD used the Pars region as the rallying and jump-off point towards the shore and the ships. Hence Parsi. It is a fallacy that it refers to Farsi without being able to pronounce "f" and hence became Parsi.
These are distinct from the Iranian Zoroastrians who came over mainly overland much later over the past 200 to 300 years, and mainly are in the business of bakeries, cafes, eateries, etc. in India and Pakistan. They are called the Irani Zoroastrians, or as we lovingly call them, Iroon for short.
Since we've been in India a whole lot longer, we speak Gujarati as our mother tongue.
While the Iroons speak Dari.
Of course we also speak Hindi, as well as the regional state language (most cases, Marathi).
But we are the same, worship in the same Agyaris and Atash Behrams, and part of the same community. And marry among each other as well.
Their calendar though is different, as are the dates of some of the festivals.