Is Social Security considered income for a parent living with me?

No, the Social Security benefits received are not counted as their income.  However, the benefits would count for the amount of support they provided for themselves. 

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 $4,300 (social security does not count) in 2021
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.

However, if your parent has other income, then some of the Social Security may be taxable.

Likewise, if you are asking about some State tax benefit where "household income" is counted, it might be applicable.