I think the question must have been asked just opposite. Means in stead of " Does a parent chain inherit the same fundamental properties of forked chain? " it should have been asked like " Does a forked chain inherit the same fundamental properties of parent chain? ".
A forked chain definitely a inherits the same properties of parent chain. But the different parameters and rule can also be adjusted in forked chain. But the the core fundamentals remain the same in the forked chain as that of parent chain.