Sorry to sound harsh but dads have got rights too and if he doesn't want to miss his child for two weeks then that's his right.
Yes, he does have rights. So to exercise those rights in this case, he has a choice. He takes time off to look after the child thus giving you the time to go and support your mum or he realises he needs to let you and his child go and do this.
And, anyway, who on earth puts a time scale on someone's health! 'I don't want you to be away for more than x number of days' sounds very 19th century to me. Most people would say 'of course you go. Let me know what I can do and when to expect you back' and that'd be that.
My mum had a heart attack six years ago. My husband was working away elsewhere in the UK at the time. He came back, picked our son up from our neighbours and had arranged to work from home by the time his plane touched down (he was working in Scotland, we live in South East England). Because he knew I'd be needed elsewhere.
Now it may be your aunt and you may not be close but her children need someone to look after them. Your mum can't do it so you're stepping up to the plate and your DH has to step up alongside you.
With luck it won't be a 2 week stint and your aunt will be in a more stable position soon. My mum was out in 2 weeks but, by then, she was well and our visits weren't daily - we shared the days between us so we did a visit every three days as an individual. Obviously you're aunt may be in for longer and when these things happen it can take a few days for everyone to find their pattern. Once you have the pattern sorted out, friends step up, neighbours step in, parents at the school may lend a hand in feeding kids or school pick up and a routine can begin which may need you to only go for a few days a week. But for now you're needed. That overrides 'parents rights' - you're not exactly leaving for good for goodness sake!
And what the hell's wrong with him travelling down at the weekends? Lending a hand. Giving you a break. Seems a bit me, me, me from him at the moment particularly when you gave up time to help his family when, actually, he should have been doing it.
I hope your aunt soon recovers and I hope your DH finally mans up a bit. Your mum is lucky to have you. Look after yourself.