(Took the question to be for what would I do 'with them', rather than within myself.)
I think about this quite a bit because I'm somewhat morbid. With my mum, I'd be absolutely stunned. I would spend tons of time with her, I'd treat her all I could. I just couldn't imagine it with her though; she's such an alive person and in the past ten years especially and since her divorce from my father, has really started living 'for herself'. I just don't know what it would do to her, that kind of knowledge, and I really don't know how I would respond around her honestly. We don't say 'I love you' and such - I'm not that kind of person and neither she is, I would find it awkward. We are close but in a very realistic somewhat strange manner. She hates things like Mother's day and is generally quite cynical about unnecessary displays of affection, particularly if commercial. So it would be a strange thing to approach with her.
Different with Robin, my father. We don't get on too well, he's exhausting to be around and I'd just rather... not. So I would suck all that up if he had this kind of news. I would humour all of his irritating, ignorant comments and his sometimes repulsives mannerisms... I'd just put up with it and focus on the good things, because he is a bit of a lonely guy (lives between France and UK, backstory blah) and loves to be in the company of his daughters/most people.
I think it's just about doing what they want, giving them what they need and sacrificing your own needs if need be. I would outright ask them what they wanted from me. I would also voice record many conversations, and start specific conversations about their histories, my childhood, family stories etc. Sorry for your situation, words can't really say how tricky it must be to know what to do.