Transferring PhD programs due to parent's sickness
Advisors are humans too (at least most of them). Most would be willing to think with you to find creative solutions. Maybe you can stay with your old institution, but arrange for you to visit the new institution. I my old institution a couple of months in another institution was mandatory. So you can keep the progress you have made and still spent some time closer to your family. Maybe, you and your current advisor can arrange for someone from the other institution to become your co-advisor. This might enable you to regularly travel back and forth as part of your normal work, and combine that with family visits. 

If people are willing a lot is possible, and in situations like yours a lot of people are willing. So my advise is talk to your advisor. Don't limit yourself to that new institution. Your primary aim is to spent time with your family, and the new institution was one possible way of achieving that. Maybe your advisor has other ideas on how to achieve your aim that work out better.