Refinance program to help elderly parents

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Our lender has a program that allows children to help out elderly parents with their mortgage through a refinance process. There are some guidelines for "elderly" but so long as we can show underwriting that our parents meet these guidelines, then the lender would allow the child to take over the remaining balance of the loan. Title, deed, etc. will be under the child and the parents can continue living under the property. Of course, the child must also qualify with the lender first.

My concern about this process is how it affects gifting and the cost basis of the home for the child. For example, let's say for simplicity, that the cost basis of the home for the parents is $100k and they have a balance of $50k. The child qualifies for the $50k refinance and the parents meet the "elderly" guidelines. The fair market value of the house is $150k at the time of refinance/acquisition. The child assumes the mortgage, title, deed, etc. to the home. 

Does this mean that the parents just gifted the child $50k (or $100k per fair market value)? 

Is the cost basis of the home for the child $50k (refinance cost), $100k (parents' cost basis), or $150k (fair market value)?