It sounds like the lady is getting SSI and SSDI confused. When you recieve SSI, and living with your parents yes you are supposed to contribute to bills. It's one reason they give it to you. Other wise if they don't think you are paying a "fair share" in contributing to the bills, they can take away a portion of the SSI.
I understand if the lady is saying your SSDI is supposed to go towards bills, because it is. But are they telling you the check will be in your dad's name? Or that you have to show proof of helping with bills? But regardless the amount of SSDI is not contingent on what you give to bills, it's how much you have worked in the past.
Depending on how your parents feel about it. You can always go get those receipt books from any store, like wal mart or target..and start making a monthly receipt. Saying you have paid your parents. Whether you actually do or not is up to you. Your parents could take that money and turn around and give it back in food or whatever else you need anyway...so you can do that or just keep it. Plus it will help them out a tad bit with other bills if need be. If you feel this is to "Crooked", then do something at the house as payment so you can keep that portion. Like extra chores or something. Something that you can do when you feel well.
Are you getting both SSDI and SSI?...I don't know how common that is but I have heard of some people getting both...and that would make more sense as to why the lady was saying you had to give a portion of your check to your father. Or are you drawing SSDI from your fathers work history?