If you pay of the lien you will basically provide a gift to your parents. You do not gain an ownership interest by paying off a debt.
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The only reliable means of obtaining ownership of your parents home is for them to convey it to you by a deed. Merely paying off liens does not vest ownership in you and it does not satisfy the mortgage. If your parents desire to convey the home to you, then they and you should meet with a real estate attorney to accomplish this and to address the legal issues that would accompany such transfer, including: 1) you paying off their mortgage by paying cash or by obtaining a new mortgage in your name; 2) investigating the liens and whether they attach to the home; 3) where your parents will go.