DH and I have delivered political leaflets on a huge ex-council estate on the outskirts of Manchester for about 15 years. The area has never been regarded as the best but honestly the people we've met along the route have all been lovely, with hopes for the future, disciplined kids with good manners and so on. Normal, in fact. Yes, lots have problems (including brushes with the law) and some houses are very run down (even smelly or boarded up) but lots more are neat and smart. Tons of the residents have lived there for donkeys years and wouldn't consider anywhere else to live.
House prices there have followed all national trends. I'd happily live there if I wasn't well off enough to live in my 5-bed detached in a country town a few miles away. Maybe I'm more "flexible" than your parents cos I was brought up in a poor but hard-working family with most of my friends living on the council estate very similar in the 1960s to those where I deliver my political leaflets. Maybe your parents have never experienced living in this sort of area. And believe me I understand how those who've never lived in such places can feel so I'm not about to chastise them for that. Your DS probably already mixes with children from such estates and some might be his besties.
So whatever offer your parents made to you, and then withdrew, was misguided, and ill-explained, but there we are. It is not your fault, nor theirs, just a mistake and not grounds for chastising them or feeling anything negative about yourself - just very disappointed. Others on here have suggested ways forward on your own or with their temporary help, so consider those and get some appointments to discuss with mortgage brokers or lenders directly. There are all sorts of mortgages out there nowadays.
I also had ten years of shared ownership on a one-bed flat when I worked a long way from home. It wasn't a great area, on the outskirts of Derby, but it trebled in price in those ten years. The flat next to mine was bought by a young single woman who installed a new kitchen and bathroom at a cost of £20k and promptly sold it for £50k more than she had paid for it!
Point out these two scenarios to your parents and let them think on it for a bit - they might change their minds. Also maybe take them along to talk together with a mortgage adviser about their ideas so both you and they are better informed about possibilities.
Don't give up hope, keep looking, planning, saving. Tell yourself you will get there. You sound like a fantastic daughter and mother.
Meg