Some additional information would help in answering your question. How much can you sell your current house for? When you say someone is interested in you house and can pay $70 - 80,000, is that total or is that the down-payment on a land contract? Will your cousin sell to you on a land contract? What do you mean by “carry the remaining balance over to a new mortgage”?
If you sell your house to someone on a land contract, that normally is a “sale” under the standard “due-on-sale”clause in mortgages. If you sell on a land contract, the lender may or may not chose to exercise its “due-on-sale” clause. If you do not pay off your current mortgage and you apply for another mortgage to purchase your parents’ house, the lender will take the existing debt into consideration when deciding whether or not to approve the new mortgage. That my or may not be a problem for you depending on other factors.
If you owe $122,000 on your current mortgage and can only sale your house for $70 - 80,000, you have no equity in the house will have to see if the lender will agree to discharge its mortgage and allow the sale even if they receive less than the amount owed. That is a “short sale”. As part of a short sale, the lender may or may not agree to accept the sales proceeds as payment for the larger amount owed. A discharge of a mortgage just means that the lender will release their mortgage and allow the sale to close. They may still pursue the borrower for the difference - the “deficiency” or they may agree to settle for the lesser amount and waive the deficiency.