My parents own a mobile home in Illinois and rent the lot. The park owner is requiring that they repair a buckled walkway due to the roots of a large tree located on the lot. Whose responsibility is it to maintain this?

Who put the walkway in? Its unlikely it was there when they moved the trailer there as trailers don't all have their doors at the same spot.

Park owner owns the lot. Unless their contract indicates they would be liable for those costs, I would tell the owner to do it.

Is the park owner claiming that the damage to the walk is somehow the fault of your parents?