You are correct, the relative clause is misplaced. It should be placed thus:
Peter, whose parents were from Japan and India, was born in England in 1982.
To maintain the given sequence of topics, you need to break it into coordinated clauses; the best coordinator here would be a point, either a semicolon or a period.
Peter was born in England in 1982; his parents were from Japan and India.
You can also manipulate the content a bit to make the fact in the second clause a second adjunct:
Peter was born in England in 1982, of parents from Japan and India.
I also applaud MunchyWilly's second solution, relativizing the first clause instead of the second.