Can we select someone in our family to be the primary caregiver and get paid for providing care?

You should get this done legally through an attorney so there are no questions down the road. Also, the amount would depend on costs in your area. But, yes, it can be done in most states if the elder has the money. One reason to get this done legally, is because if the parent runs out of money and needs to go on Medicaid, all expenditures will be looked at. So, see an attorney who knows Medicaid laws to do it right.
Carol