MUMBAI—A battle to control India’s largest global conglomerate, the Tata Group, is convulsing the powerful ethnic Parsi community, whose club of billionaires has played an outsize role in the nation’s economic evolution.

The Tata Group—which sells more than $100 billion a year of everything from Jaguar Land Rovers to Tetley Tea to rooms at the famed Pierre Hotel in New York—is engaged in an unusual public spat between its ousted chairman and his retired predecessor. Both men hail from India’s Parsi community—a group of about...