K.P.
You can only do this if you the provider is "claiming" the income on their taxes. When you submit your reimbursement forms, they will ask for either the provider's Tax ID or SSN and the government will be looking for that income on their returns!
I do believe that $5000 is the max you can claim in childcare expenses, so you would be "giving up" the credits.
Yes, this will raise eyebrows on several levels. If your parents are being paid and you are using an FSA, then they need to claim the income on their taxes. If they then elect to "regift" that money- they need to open an account for your son in their name as a "trust account".
Really, are they charging you? If you're not paying for childcare, this is kind of irrelevant. Just know that your providers could easily be audited if you are claiming something that isn't true.