It's best not to put all of your eggs in one basket. If probably is not advisable to assume the bank WILL allow your parents to remain after foreclosure and continue paying rent.
Our recommendation is to explore other options where they will be able to enjoy much less uncertainty.
Most banks will allow the tenant to continue to stay until the home is sold, My opinion is your parents need to start looking for another place to rent as soon as possible so they are not forced into making a quick move when the home is re-sold. This is starting to be all too common, landlords not telling the tenants they have not been paying the mortgage on the home. Then tenants finding out the hard way they have to move. I am not an attorney and do not know what your parentâ€™s rights would be so they may want to talk with an attorney.