My elderly parents have put my name on their savings account. I am filling out a FAFSA form, do I have to claim their savings?

It isn't likely that you have to claim the account. Generally, when there is more than one name on an account one person is considered the primary account holder and their social security number is listed. This doesn't actually mean that one of the joint account holders is more in charge or has more claim on the money - it just means that for interest baring accounts the IRS needs one person to assign the income to.

If you were added to the account after it already existing for a time, it is most likely already listed under one of your parents SS#. 

Presumably your parents have put your name on their savings account so that, if necessary, you can access THEIR money for THEIR use. They haven't just given you the money!

So it's not yours, so I'd say no. But I am ready to be corrected?

You only report income on FAFA so in 2015 who claimed who on taxes? If they claimed you you need to use their AGI as the household income amount" if your parents have you on their savings that is not the same as income, so you don't need include savings.