Ok, just to clear a couple of things up.

The amount of money you owe the bank is the "loan" or the "debt".
The "mortgage" refers to the legal instrument that the bank holds over your property, it's basically the document that the bank uses to establish the property as "security" against the debt (their security, not yours).
So with the guarantee, the bank will draw up a (probably limited) mortgage against your parents, as security against the event that both you and your parents are unable to pay off the debt.
"Equity" by the way refers to the amount of (not encumbered by debt) value in a property. So if the real value of the property was $500K, and you had a loan on it for $200K, then there is $300 "equity" still in the property. So if you parent's house is worth $500K and there is no debt on it, then the equity in it $500K. EXCEPT the bank valuation is usually lower than the market price to allow for variations in the market.