We see in the Hollywood movie Super Mario Bros. that no-one seems to know Mario's full name or his background.
So I just want to know, who are his parents and where do they live? Are they still alive?
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The answers really depends on which canon you use.
(Disclaimer: I have never watched the Super Mario Bros. movie.)
We don't know if their parents exist in this canon.
Their parents do exist (see Yoshi's Island) but don't play a large role in the games.
The parents live together in a small cottage in the hills, probably near the Mushroom Kingdom. Not sure if they are still alive.
Their mother definitely exists and lives in New York somewhere, and is very much still alive. We don't know about their father.
The only story that has Mario's parents is Issun-bōshi. They live in a cottage near a river, and are probably still alive at the end of the story.