How to help my mum to eat more and heal relationship with food

My mum (she’s 62) has always had a strange relationship with food, but it’s got worse in the last 10 years I would say. She was always the mum who put dried fruit in our lunches instead of crisps, wholemeal bread instead of white bread (don’t get me wrong, I’m thankful she helped me have a ‘healthy’ childhood! But that’s probably my earliest memory of her seeing certain foods as ‘good’ or ‘bad’).

We’re a slight family, all what you’d call ‘slim’ but she is very skinny, and I’m worried (and have been for a long time). She used to eat meat but when I went vegetarian 10 years ago (I now eat flexitarian), she pretty much went vegan, and says she could never eat meat again. She mainly eats beans/pulses and vegetables, no ‘junk’ food whatsoever - and kind of shames my dad (who has a very healthy relationship with food) for enjoying things like fish and chips, macaroni cheese etc.

I am worried for her health as she gets older, she’s so weak already and is only in her early 60s. We’ve tried to encourage her to eat more but she never sticks to habits and sees fatty or ‘indulgent’ foods to be associated with guilt (e.g. ‘I can’t have a cake because I had a piece of chocolate yesterday’).

For context, her mum/my grandmother is classed as ‘obese’, so unsure if that has something to do with her not wanting to end up the same way?

Any tips for helping her to eat more and heal her relationship with food would be massively appreciated, thank you :)