Oh OP I am so sorry at this turn of events.
Firstly you are NOT a failure in any way. You are a survivor and you should be commended for that. Others perceptions, and mis-perceptions do not change that.
Second your parents plan for them to completely own the new, second house is wrong in so many ways - why would you want to pay a mortgage for something you don't own? How would you even get a mortgage if you have no stake to stand as collateral for that? And if your parents did need care in the future whilst the family home would not be seen as assets whilst the remaining spouse lives there your home, fully owned by them most certainly would be - you would risk the home being sold to pay care costs and leaving, again, with nothing.
It's batshit plan that I think reflects your parents slightly strange, and I would say slightly naive attitude to money - just because they've accumulated a large pot of savings doesn't mean they are money savvy, just very frugal, it's not the same thing. They seem to need to "keep" their money even when they're spending it, hence "buying" you a house but wanting it in their name.
You are again displaying such kindness to them in not plainly saying to them how much of mess this plan has created but don't think your feelings don't count, you are allowed to express those too.
I do think you need to be very wary of any further "offers" of help - your parents seem to have very particular ideas of how their money should be....used, and for them to be offering what the rest of us see as an obvious choice (gift you the money for a deposit no strings attached) would represent a sea change in their attitude to money. I don't think they are going to have that.
Deep breath, back to the drawing board and perhaps a little distance from your parents and their "help" would be a good thing right now.