Can't find anything about his parents/family.
Found something interesting, though…
In 2009, Cook said that he offered a portion of his liver to Jobs, since both share a rare blood type. Cook claims that Jobs responded by yelling, "I'll never let you do that. I'll never do that."
https://en.wikipedia.org/wiki/Tim_Cook#Personal_life
https://www.washingtonpost.com/news/morning-mix/wp/2015/03/13/apple-ceo-tim-cook-offered-steve-jobs-a-life-saving-gift-but-he-refused/?utm_term=.a70558ef3830
Now, what rare blood type could that be…