Unfortunately the advice that your parents received did not account for this contingency. It is true that legally your parents have the right to assign or encumber their interest in the property. The problem is that their interest in the property is extremely limited and not many lenders are willing to take the risk.
The previous answer suggested that you might seek to put your name on the loan. That may be the only way the bank will approve the line of credit.
Additionally, if your parents' health is declining you should make sure that their estate planning needs are taken care of while they still have the capacity to make decisions and make any necessary changes. You should consider not just the Will, but the Living Will, Health Care Surrogate, any trusts they may have, bank accounts, life insurance policies and other assets. Your Power of Attorney is only valid as long as they are alive and you will have no authority to act once they pass.
An experienced Estate lawyer can help you with these issues.
You may be legally able to obtain a hole equity loan but that does not mean that the bank has to grant the loan. The banks have their loan underwriting requirements and that may be the problem. Will Bank of America provide the home equity loan if you are also on the mortgage? You may also want to check with other banks to see if they will do it.
No attorney-client relationship is created by answering questions in this public forum. If you wish to create an attorney-client relationship, you must contact me directly, not through this public forum. Answers are provided based on general ideas and an answer specific to your situation would require a review of all documents.