I have no idea about my father's ticklishness, or anyone in his family. As for my mom, I only tickled her once in my life, as I recall, she was moderately ticklish.
As Iam very to extremely ticklish, I think that I probably get it from my maternal grandmother, and one of my aunts, both of whom are/were extremely ticklish. When I was younger, like pre teen years, I tickled my aunt's feet once or twice, and she went ballistic. I recall my grandmother telling me once that if my grandfather so much as accidently touched her foot while in bed, say with his foot, it would make her jump a mile in the air. She claimed to be extremely ticklish, even in her 70s.
So, for me, I would say I probably get my ticklishness from my maternal grandmother, and one of my maternal aunts.
Mitch