You are not being unreasonable gilly. You sister is being thoughtless, as is your Dad who could have texted your sister to see if she could take him shopping.
Your post struck a chord with me because when my elderly Mum was alive, all the work, caring, outings, fixing things, Drs appointments, shopping, company etc, etc was down to me as my siblings lived further away. I was the only one with no husband (divorced) a full time job, young children and a life full of worry. Looking after my Mum was a labour of love, because she was lovely and appreciative of my help (and apologetic too because she felt she was a burden - she wasn't) but my brother and sisters let me get on with it. They thought they were doing me a favour if they came home at the weekends to see Mum for a cup of tea. They really thought that was sharing the load! I used to fume too.
It seems the more you do, the more you are expected to do gilly. It is good that you are capable but you could do with a break too. Some people have a thick skin and no empathy or conscience it seems.