Can a parent be claimed as a dependent for taxes if they receive SSI?

No one can tell you if your parents would be your qualifying relative (QR) dependent on your income tax return. You have to decide this on your own because you and they have all of the necessary information that will be needed for this purpose. You can claim your parents as a dependent if they meet the definition of a qualifying relative. This requires that the parents be a US citizen or resident of the US, Canada or Mexico, they are not filing a joint return other than to claim a full refund of any tax withheld, their taxable gross income was under $3650 for each one and you provided more than 50% of each ones total support.

The individual DID NOT PROVIDE MORE than half of the individual total support for the tax year. The cost of support includes food, clothing, education, medical and dental care as well as the fair rental value of shelter provided. This must exceed the total of all other assistance coming from other sources, including other relatives, government entities, the person's savings and non-taxable income.

Go to the IRS gov web site and use the search box for the referenced material.

References:

Publication 501, Exemptions, Standard Deduction, and Filing Information

Tax Topic 354, Dependents

For a worksheet to use for determining support go to the search box at the IRS gov web site and refer to publication 17 go to page 37.