Hi - my name is XXXXX XXXXX I'm a Bankruptcy litigation attorney. Thanks for your question, but I'm sorry you and your parents are having to go through this ordeal. I'll be glad to help you here.
I'll give you a little background to make this more understandable. When a person/couple (debtor) files bankruptcy, all creditors are automatically stopped from pursing the debt for payment of the debt, recovery of the collateral, etc. - called the "automatic stay".
HOWEVER, secured creditors (lenders with loans secured by property - house, car, etc.) have the right to file a motion to lift the automatic stay and recover their collateral in order to sell it and then apply those proceeds to the debt. This is what your parents first received from the mortgage company ("Notice/Motion to Vacate Automatic Stay"). Thus, this motion asks the court to allow the creditor to re-take its collateral, and the courts are generally always going to grant the motion because the lender has a right to recover its collateral.
If no one ever responded to the motion of the mortgage company, an order was likely entered on 6/11 granting the request to lift the stay. This means that the lender - after 6/11 - has permission to proceed with foreclosing on the property.
As for how long your parents have before the lender forecloses, no one can tell you that for sure - - except the lender. There is no real time limit on foreclosing/no requirement to foreclose within a certain amount of time from the automatic stay being lifted. Thus, who knows when the lender will begin foreclosure.
But, your parents will receive notice of the foreclosure because NJ requires mortgage companies to perform a judicial foreclosure, which means that the lender will actually have to file a lawsuit to foreclose. Because of this, it usually takes at least 8-10 months before the foreclosure concludes - - and your parents would be allowed to remain in the house until the case is over and the house is sold.
Here's a very good link you can read about the process: http://www.foreclosure.com/statelaw_NJ.html
Thus, your parents should have a reasonable amount of time even after the foreclosure process starts. It will not be a situation where sheriff shows up and tells them to get their things out of the house immediately. They will have adequate time. Also, many lenders will offer compromises and even help with the cost of moving in exchange for a peaceful return of the property to the lender, etc. So, they can likely expect something like that to occur as well.
I hope this answers your question, but if you need something further, please don't hesitate to ask. Thanks.