Don't know abt Canada, but in the US (informal) mortgage assumption was done away with long back.
In all motgages there is a 'Due on sale' clause that states if a prop xfer ownership, the mort. co. has a right to call the loan.
Note: the wording says "right" to call the loan.
In the US what I would do is simply deed the property over. 99% (if not 100%) of the time the mort. co. will not call the loan as it is not in default. They have their hands full wil already defaulting loans.
Note: in the above scenario, the neighbor and not the parents will still be liable for the loan even though the parents are making the pmts.