What about a situation where the parents put up the money for the down payment, but then are partial owners? i.e. they pay the 20% down payment but then are entitled to 20% of the equity? Does such a thing exist?
You'd have to ask your bank but I highly doubt it would work. They probably wouldn't consider it a residential mortgage, because two of the owners (the parents) wouldn't be living there. You could always do it on the back end--have the parents provide the money as a gift, close on the house, and then after closing go see a lawyer and have papers drawn up to make the parents 20% owners. The downside there is that you might have to pay transfer tax (local stamp tax, paid when property changes hands) on the 20% being transferred to the parents. The other downside is that if this is your plan from the outset, then technically you're committing mortgage fraud because the parental gift wasn't really a gift.
You can't do that because the mortgage lien on the property prevents any transfer of ownership without the approval of the mortgage company, and they aren't going to approve you just handing away 20% of their collateral. Also they'll (rightly) cry foul for mortgage fraud.
No, you just have to transfer to someone who "takes title subject to the mortgage" (i.e. their interest in the property is subordinate to the bank's). Even if you don't fill out paperwork specifically saying they're taking subject to the mortgage, in most if not all states that's automatically how they'll take their interest, because banks just about always record the mortgage and recording it functions legally as if they had told everyone, including future buyers or future people who take a legal interest in the house, that there's a mortgage on it.
In other words if you buy a house and then turn around the next day and transfer title to, say, your husband or parents or whoever, you did not just transfer away the bank's collateral. Putting someone else's name on the title doesn't take away the bank's collateral any more than putting your name on title at closing did. The title is in whoever's name, but the bank still has a lien on the house.
The potential or theoretical fraud isn't in transferring part or all of your interest to someone else, it's in concealing from the bank that what your parents intended when they gave you money was not a gift, but a loan or partial interest in the house.