It is necessary to make several assumptions in answering this question. Firstly that the funding comes from Social Services and that the emergency either was in regards to your sister, or it directly affected her. We’re not sure that the emergency was simply that your parents were away for longer than the planned respite for that particular period? We will also presume that your sister is provided with a council care plan.
The Local Authority (LA) is correct in indicating that they should have been contacted in the first instance. They are ultimately responsible for the care provided and when someone’s main carers aren’t there, the council should have been informed so as to decide what to do to meet the shortfall in need.
There is no obligation on people to step up as described by the council. But if they have, then unless they have notified the council, that their action is not voluntary, it will be regarded as voluntarily provided care. The whole point of review in the Care Act is so that people can go back to the council if there’s a problem with the care plan.
The care plan should advise what to do in a real emergency, what action may be needed, however if there is no such information within the care plan.
Para 10.44 of the statutory guidance https://www.gov.uk/government/publications/care-act-statutory-guidance/care-and-support-statutory-guidance explains what should occur for someone an individual with fluctuating conditions. It indicates that the plan should make ‘comprehensive provision to accommodate for this, as well as indicate what contingencies are in place in the event of a sudden change or emergency’. Further it states that ‘this should be an integral part of the care and support planning process, and not something decided when someone reaches a crisis point…’
In a situation where more than planned for respite or intervention in an emergency was provided, then whilst there is no RIGHT to be paid for it, the circumstances should be listened to properly and a decision made on the facts. If places were theoretically available for short term residential care, in a setting that would have been conceivably suitable, then it should be for the Local Authority to prove a vacancy provision existed. If such occurred then it may be within their rights to refuse payment for the week, or at least only pay the amount it would have cost to provide the short term care.

Thank you very much that’s so helpful. The emergency was my father was hospitalised for 2 weeks with appendicitis. My Mother couldn’t care for my sister alone so my brother had to fly out and collect my sister and then all provide care for a week in the UK until they returned.
There was no contingency plan in her care plan so I will ask the social worker about what care would have been available and make a plan going forward using the information you have provided.