I have been in a similar situation before. Personally I think it's a terrible idea to try to manage your parents' finances IF thats' going to lead to a lot of strain in the relationship. If your relationship with them is good and you can discuss options with them then I'd suggest a couple of things...
1) Get new quotes from contractors on the renovations, maybe that's something you should get involved in (or not!). ~600k on renovations is insane. Sorry but that's my instinctive response. Talk to your parents about paring down the wishlist while you're at it. Also where are your parents going to live while that much work is going on?
2) You and your sister should offer a lump sum first. Then take over the mortgage, rent out the house. Room by room if you need to get higher cash flow and have time to manage. It's actually not that hard to do room by room if you screened well and look for the right kind of persons. Or just regular LTR. Send your parents the profits and then some per month.
It's an unfortunate situation for the kids. Looks like your parents essentially planned their retirement around selling the family home, so this sounds like a tough situation on many levels, not just financially. But if you really want to save the house, appeal to their sense of nostalgia and see if they will work with you on this. Best of luck.