Whoever is legally obligated to pay the debt and made the payments. If you are not on the mortgage and you don't pay mortgage interest, you can't take the deduction. Your parents can take the deduction.
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Trusts must file a Form 1041, U.S. Income Tax Return for Estates and Trusts, for each taxable year where the trust has $600 in income or the trust has a non-resident alien as a beneficiary.
If you need to file a return for the trust you will need TurboTax Business. You can visit the following link to purchase TurboTax Business for 2020: TurboTax Business
So does the trust have to file a tax return separately or does it go on my tax return because I am the grantor?
Grantor trusts can typically use an optional method of filing rather than filing a 1041.
NOTE: Also, for future reference, in order to deduct mortgage interest, a taxpayer must actually pay the interest and the taxpayer must at least be named on the deed (i.e., it is not necessary that the taxpayer who pays the mortgage interest be named or otherwise obligated on the note).