Please help -- need to buy my parents a house?

Hi all, will caveat this by saying I really do not know a lot about real estate so apologize for any obvious / stupid questions, but really appreciate any input you might have.

My parents really want to buy a house within the next year, but won't be able to do it on their own and asked for my help. The goal was always to buy my parents a house, but they've definitely raised the issue a lot quicker than I would have intended and I'm not sure what would be the best way to go about it. They're both getting pretty old (60+) and still live in the same 1B1B apartment I grew up in, so I really would like for them to finally live in a house that's ours and have their monthly payments go toward the house vs. rent. However, my parents have a troubled financial history (bad credit, bankruptcy, foreclosure, etc.) and they don't earn a substantial amount of money (dad is a general contractor, mom used to be a waitress but can't work anymore due to health reasons), so don't think they would qualify for a mortgage. I think my parent's idea was to put together some money and buy a house under my name that they would would live in, and they'd pay me rent that I would use to pay the mortgage, and eventually I'd be able to do whatever I'd want with the house. We're not looking for anything that nice -- just a small, modest home in the suburbs of CA. 

As implied from my title, I just recently started my associate gig at a PE firm (after 2 years in IB) so I'm still pretty young. Currently have about $85k in savings (NOT including my retirement accounts) and am set to receive another ~$65k bonus at year-end (but a lot of that will go to taxes). I'm just not really sure what would make the most economic sense with how much I have right now and where my parents are at. My parents said they want to contribute but am I in a position to be able to do this on my own / should I do it on my own? Any tips on how to navigate this situation in general? Again, sorry for my lack of knowledge on this topic, and thanks in advance for your help.