I agree about not knowing how unwell you feel and not realising it until you're better - I was the same before I had my hysterectomy, when it was over I couldn't believe how ill I'd been feeling.
My husband won't exercise either and refuses to come with me on walks or to the swimming pool, and I've given up trying. He is Type 2 and is also very overweight. He loves the LCHF meals I cook, but then goes out and buys himself pies and burgers and chocolate - he also eats bananas which he buys himself, as I've removed all cakes and biscuits and bananas and grapes from the house - not easy when there are young people in the household though.
The big problem with LCHF is that there's no room for cheating. Its quite a high calorie regime which is why you don't feel hungry, but your Dad has got used to snacking on biscuits and this is undoing anything he has gained from low carb-ing.
Your Mum sounds as though she's exasperated with him and that's why she's given up trying, she may be depressed about him too. Please don't feel bad about this.
You obviously love your Dad and are feeling desperate but do avoid being the diet maffia, it won't help. He needs encouragement not criticism.
I think you've started well by talking about it with him.
Does he have a meter to check his blood? GPs don't usually hand them out to type 2s.
They are quite cheap to buy but the testing strips are costly. I found that testing my own bloods and comparing them with his results got him interested to start with (the 'competition element I think) - could you bring yourself to test yourself alongside him? Once the first (actually surprisingly painless) finger jab is done it's easy (but you'll use up a lot of expensive test strips if you're both doing it).
That is just one idea, along with the dog walking and other activities already suggested on here, remember to him the owrd exercise is a swearword.. You and your Dad will soon be best buddies - but please avoid any sign of 'nagging' - that was my mistake and it made him sneaky about his extras.
Think positivly, give him encouragement - you are doing great.
Little steps.