First of all, with regard to your child, it is not as straight forward as it sounds.
If you register your child in the US, then she will be British by descent. This means that she will not be able to pass her British citizenship onto her children if they are born outside of the UK. If you get her to the UK on a US passport and register her once you get to the UK, then she will be British other than by descent, so she will be able to pass her citizenship on to her children no matter where they are born. This may be something that you wish to consider, especially if you may go back to the US at some point.
With regard to your husband's application, you will be the sponsor on the form, not your parents. You can supply evidence that you will be supported by your parents, buyt you must make clear that this is going to be a temporary arrangement.
You can show savings held in the US as evidence of funds, and it will probably be a good idea to show that your husband has a good chance of getting employment soon.
Vicky