I think if your husband turns down the job, he needs to be crystal clear why he is turning it down. He should say something like, this is my dream job, I was so happy with getting this position. However, my father got a prognosis and we don't know how much time he has, and I need a job the next couple months that is flexible if/or when something happens.
It is possible that they will say, see you around. Or they may be able to say, well we still need you to start but you won't have to travel the next 2 months, or whatever. He needs to think about what he is willing to accept in order to get his dream job, which honestly doesn't happen all that often. I know this may come off as being harsh, but it's not like his health is a surprise. Also, doctors are pretty bad at knowing how much time someone has, unless they are very close to death. The father may die in the next month, or in a year, or anywhere in between or even past that. I don't know if he should hold off finding a job until everything is known about his father's situation. However it is reasonable ask if the start date of job can be delayed in order to have some quality time with Dad before starting something new and demanding, or ask if could have an unpaid leave of absence in the next year (say 2 weeks) at some point due to Dad.