Hi OP
I had a similar situation. I met my dp this time last year. In fact, we were supposed to meet this week a year ago, but he cancelled our first date because his dad went into hospital.
I am not going to lie, i wasnt sure he was being truthful for the first few weeks as he was essentially a stranger (OLD, speaking for 3 weeks before we met). Well. He wasnt lying. His poor DF died this March.
I never let on i didnt believe him, but i think he sensed it must sound a strange co-incidence as he kept shuffling our dates around to juggle hospital visits with family, and i did after a month or so get photos from the hospital of crap food or whatever.. .
It was and remains tough. There were certainly times i thought i would lose DP through the stress and then grief (My DP didnt expect his DF to die, he wouldnt accept it right until it happened), and would have understood if he had left me. I even opened the door for him to leave, a few times (Out of insecurity I guess) but we battled on.
Your instincts just to 'be there' are right, and if he likes you he will continue to see you.
If he likes you he will continue to show you and tell you. I was in knots, feeling selfish, worrying about losing this guy i really liked (And for the first 3-4 months my DP wasnt that stressed about his DF as he thought he was getting better, just the visiting hospital and caring for his mum to stress about). But even when his DF died he didnt stop telling me he loved me and how lucky he was to have such a loving and supportive gf. But i honestly was crapping myself for months.
Even now, it can be hard. Not everything is back to normal. (Sex drive, for example, is nil for him still) but there is affection, close-ness, loving words.
To be completely clear too, there has never been any abuse, bad behaviour towards me, cheating or really... anything. If he starts any of that, walk away.
Of course there were some highly apologetic date re-arrangements, but that was understandable.
Good luck op 