My parents have CD's in both of their names. If one has to go to a Nursing Home will the money stay in their accounts?

As long as there is joint ownership the CDs stay in place until they mature. Then you have to decide how to reinvest. As long as both parents are living and have joint assets there shouldn't be a change unless they make one.
Are you asking because there is the possibility of your parent entering Medicaid nursing homes? There is a 5 year look back period on personal assets. You may want to look into Living Trusts.
Lilli