There weren't really any major 'Persian' invasions of India. The closest would be the Mughal invasions (the Mughals being Persianised Turco-Mongols), which were

There weren't really any major 'Persian' invasions of India. The closest would be the Mughal invasions (the Mughals being Persianised Turco-Mongols), which were relatively recent, in the history of India (16th century A.D.) There are claims of earlier 'Aryan invasions' of India (nothing to do with Persians), meaning essentially an invasion of Sanskrit speakers into India, though to what degree these migrations were 'invasions' in any sense is unclear. It is also unclear whether migrations/invasions of these Sanskrit speakers displaced Dravidian speakers. It is clear that both Sanskrit/Indo-Aryan speakers and Dravidian speakers partially displaced earlier Austro-Asiatic (Munda) speakers, who themselves seemed to have displaced yet earlier populations.