I'll be honest, this set up sounds like a recipe for disaster.
If your siblings "don't want to get involved" now when your parents are alive - what will it be like when one or both of your parents passes away?
When it comes to families, I believe lending money is guaranteed to end in tears. Give what you can afford, if you ever get it back that's a bonus. If you can't afford to give the money then you can't afford to lend it either.
As a sibling, and with the best will in the world, I wouldn't be prepared to sign a document such as you've described; it's way too open ended, you've no idea what amounts might be involved, what costs for the carer are actually included - annual leave? sick pay? overtime? travel expenses?
As elcato has pointed out, you can only do this for one year, then what? Heaven forbid, but your father could die first and then in truth it would be very difficult for anyone to keep your mum at home.
I'd urge a re-think.