A “gift of equity” refers to a gift provided by the seller of a property to the buyer. The gift represents a portion of the seller's equity in the property, and is transferred to the buyer as a credit in the transaction.
Generally, you can give a gift of equity to someone if you're a family member (which includes legal guardians), engaged to the recipient or a domestic partner. Loans backed by the federal government may restrict giving gifts of equity to family members only.
Yes, if you have a mortgage on the property your lender will need to consent to any other parties becoming legal owners. They will require the incoming party to become a party to the mortgage as well.
Keep in mind, there is no limit to how much equity can be gifted. So, if your parents could sell you that same home for $264,000, you'd have $66,000 in gifted equity. That equates to an even more attractive 20% down payment. The more you're able to put down, the less mortgage you need to take out.