Hi. I have had a lot of experience of this, having cared for my own parents, 2 neighbours and currently my PIL. Stroke, dementia,incontinence, clutter, the lot.
Please talk to Age UK. They will give you good advice.
You also need to sort out attendance allowance for both your parents if you haven't already done it.
You may be entitled to carers allowance for yourself. Ask AGE UK. this will help with the cost of a carer.
Have they got a social worker? Who organised the day centre?
You should contact social services, citing "emigration of primary carer", and ask for an assessment with a view to getting someone to do the tasks your sister was previously doing.
Social services will probably tell you that you can't have any assessment or help unless your parents consent to full financial disclosure. This is incorrect. As long as you agree to pay for the carer, this is not compulsory.
You need, as a matter of urgency, to get both your parents to sign a letter giving permission for their GP/hospital to discuss their medical history/care with you. Ensure that an original goes into their medical records, and another original is with you. This will save you endless hassle and stress further down the line.
Then you need to look into power of attorney. This is expensive, but again, will save a lot of stress later.
Clutter is a fire hazard, a falls hazard, and makes the symptoms of dementia worse, so whatever you can do to reduce that, the better. (However, I am 2 years into the process with PIL and not anywhere near winning yet).
It sounds daunting, but the sooner you get the process going, the better.
Good luck.