Hi Jackie
I feel for you - 61 is an awfully young age to be advanced with dementia.
One of the things that goes is spatial awareness, and it may be that this is what causes your Mum's falls [rather than a loss of use of limbs, which may come later].
This does lead to a loss of mobility, possibly injury.
I found that for my Jan, who was around the same age, her 'radius of safety' became smaller and smaller as her illness developed. Her lack of confidence gradually meant that she was only happy being in our lounge.
Generally I have found that dementia patients are amazingly mobile, even at quite an advanced stage. Some ricochet around Jan's care home, push furniture around, lead each other around, etc. They are okay to do that since the floors are level and furniture is safely placed. At home they might not be so safe.
There may be things your Dad can do to make things safer in the short term for your Mum. Perhaps move some furniture, or get rid of it. Also always to be on hand to help your Mum out of the car, though I know that can be difficult to manage safely, alone.