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Since your name is XXXXX XXXXX account, 1/3 of it is presumed to be yours and you would pay tax on 1/3 of the interest earned.
You do this on form 1040 schedule B
That's the only tax that's due.
However, if the value of the account is over $10,000 you will have to file form TDF 90-22.1
When they pass away, there is no tax on the funds that you receive.
But, if they pass and the value of the funds you receive is over $100,000 you will need to file form 3520.
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Also, as to the interest earned on the account, it could be a more complex tax question. The owner of the income is determined by local law. If local law pins your parents as the sole owners of the account, then you have no income to report from that account.
You can read more about how interest is allocated to accounts, here: http://www.woodllp.com/Publications/Articles/pdf/Who_Pays_Tax_On_Joint_Bank_Accounts.pdf
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