What I'm hearing is: Your father pays rent on your mother's house and she let's you keep the rent money to pay you for her living in your house.
As Carl said, this can get complicated and professional tax advice may be advised.
That said, the simple version is: Your mother has rental income and should file schedule E to report the income and deduct the expenses, particularly deprecation. You do not, most likely, have any income to report. Your mother is just reimbursing you for her share of your home expenses.
