Has anyone been in a situation where social services have made an assessment that their parent is a danger to themselves and must go into a care home against their will.
Adult (social) services have no lawful means to force her into care home against her wishes, you said she is "mentally okay" so she gets to decide to stay on her own home.
Your DM can choose to live at home with high falls risk and multiple hospital admissions and choose not to listen to social workers,OTs, her GP, her husband (DF who is her main informal carer) nor you as her family.
So there won't be any legal means to make her decide a 'safer' option that would provide 24 hour care not reliant on dad nor family, as she already has 4 x daily care package.
Unfortunately from your POV all you can do is encourage her and decide what you can cope with and do, and what not to get drawn into. If mum has Parkinson's disease then, over time that can affect cognition and cause a dementia type issue once really progressed, but it doesn't sound like it is yet at level is affecting her capacity for decision making (under Mental capacity act 2005) because legislation allows for "unwise" decisions.
It's a huge deal to override someone expressed wishes when they lack capacity even and has to be 'bullet proof' and shown to absolutely be in their best interests (not others best interests even if a strain on family). The bar is set so high as it is about Human Rights to autonomy and freedom.
The thing to remember here is that Mum wants to live at home and accepts she will fall, and doesn't mitigate those risks herself. She may well come to a crisis one day and fall and seriously injure herself, when she has less options, but for Mum it is more important to be at home despite those risks.
Dad and you can only do what you can and accept this is important to mum. Please don't feel guilty if you aren't available sometimes. It's her choice and yours for when you can help and when you need to sleep or give priority to your own family and need for a break from being endlessly on call. Dad can call an ambulance. GP can talk to her after next crisis.
You'll find social workers sympathetic to your frustration. They may ask but actually don't expect, if they fully understand your perspective.