The visa form for US residents specifically asks for the name of my husband or father:
Other countries (e.g. France) ask for detailed information about both of my parents.
Why does the US form ask for "father or husband"?
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I don't know why India asks for it - but I do know that in some countries the name of the guardian (either the husband for women or father for males and unmarried girls) is part of your passport identification page.
This is checked when you are traveling - for example, if I am to sponsor my wife's visa, her passport must state my name under "Father/Husband", otherwise the application is not approved.
It might be something similar in India.
@choster's guess about the name is also a good possibility.
I think this is quite common in some Asia countries. For example here in Thailand it is the same, even though my parents are now long dead I have to list them on various forms.
In the case of India I think they love bureaucracy, it is an old habit and keeps lots of people in jobs I guess. Blame it on the Brits I suppose, though overall I think they helped.
I think you looked at the first page only. The first question on the second page of the form you linked to is
Name and nationality of mother.