If your dad says yes and you say no because you're in fear of ruining a relationship that is going fine right now, you're going to seem like a complete jackass. If one of your boys had two dying parents, you'd probably stick up one for them. The relationship should be the last of your worries right now, especially if someone you know is going through a hard time.
I don't see how rejecting her when she has absolutely nowhere to go is going to help the relationship. What would you think if you were in her situation? Both your parents are dying, you need a place to stay, your girlfriend's parents okay it, but she says no.
That would ruin my day.
Why don't you quit wondering about her. It already seems that you are more obsessed than she is. You think too much about things you can't control.
"Is she going to get tired of me" is something that you shouldn't be worried about. "Are you going to get tired of her" is the type of thing you can control and figure out on your own. Things are going to be different, but if you can't manage through this, I don't see how "long" you can make it.