If my elderly father lives with me, can I claim him as a dependent on my shirt form taxes?

You may be able to claim him as a dependent on your tax return using the Qualifying Relative rules if he meets all the requirements under the rules.

To be a Qualifying Relative -

1. The person cannot be your qualifying child or the qualifying child of any other taxpayer. A child is not the qualifying child of any other taxpayer if the child's parent (or any other person for whom the child is defined as a qualifying child) is not required to file an income tax return or files an income tax return only to get a refund on income tax withheld.
2. The person either (a) must be related to you or (b) must live with you all year as a member of your household.
3. The person's gross income for the year must be less than $5,050 (social security does not count) in 2024
4. You must provide more than half of the person's total support for the year.
5. The person must be a U.S. citizen or a U.S., Canada, or Mexico resident for some part of the year.
6. The person must not file a joint return with their spouse with the following exception -
You can claim a person as a dependent who files a joint return if that person and that person’s spouse file the joint return only to claim a refund of income tax withheld or estimated tax paid.

Maybe.  See the following TurboTax help article on:  Who can I claim as my dependent?