Relative clause with "whose": is it misplaced?

Whose is the right relative pronoun to use, but the placement of the relative clause makes it awkward and incorrect. Relative clauses typically closely follow their referents:

Peter, whose parents were from Japan and India, was born in England in 1982.

Alternatively, you could try to subordinate the other clause like so:

Peter, who was born in England in 1982, had parents from Japan and India.

Unfortunately the genitive relationship between Peter and his parents doesn't fare well across the relative clause (as in your original sentence as well), so a different construction must be used if you wish to maintain the initial ordering of the clauses.

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.

ADDED:
I also applaud MunchyWilly's second solution, relativizing the first clause instead of the second.