It seems very unlikely that "78% of black kids had parents who grew up in the top [60%] and moved down at least one."

Perhaps it should have been written:

25% of black kids [who] had parents [that] grew up in the bottom two levels [] moved up at least one [but] 59% of [such] white kids moved up.

78% of black kids [who] had parents [that] grew up in the top three levels [] moved down at least one [but] 43% of [such] white kids moved down.

But I agree it's unclear and this is just speculation.