A babysitter does not provide receipts. However, I think they are still expected to claim their income (like servers are with their tips) but I'm not 100% sure on this.
I would think it would be hard to find a handful of parents who would agree to not getting any receipts so it wouldn't be as viable as just calling yourself a hdc and taking the legal steps for doing so.
If one is a hdcp already, no they can not take a child "under the table" and have different rules for that one family