I can only speak from anecdote and personal experience with regard to this question. (Looking forward to the day when I have the resources to do real research in this area!)
I don't think that parents' seasons have no effect on kids' seasons. Since skin tone is heritable, it makes sense that season would be heritable.
But I don't believe we can predict with great accuracy a child's season from a parent's season-- just as we can't predict with great accuracy a child's hair or eye or skin color based on the parent's.
We can make good guesses-- two Winters are probably likelier than average to have a kid who can wear black, don't you think? -- but we can't know for sure.
I don't see clear patterns in the families I know. In my own family of origin, my mother and father -- a (probable) Bright Winter and a Soft Summer -- produced a Dark Winter daughter, which you might very well expect, and a Soft Autumn daughter, which you probably would not expect.
In my family of generation, as they say, I combined my Soft Autumn genes with those of a Light Summer. We made two Light Springs. They're warm, like me, but they're more saturated than either of us. Who would have predicted it?
What combinations do you see in your family? Share in the comments! :-)