Is it permissible for me to arbitrate between my parents who are disputing over a plot of land and a house, due to the absence of Sharia courts and their desire not to involve an outsider, and would I be considered undutiful if I asked one or both of them to swear by Allah regarding what they said?

There is no impediment to your arbitrating between your parents if you are qualified for it with knowledge and understanding, and if you judge by what Allah has revealed, not by man-made laws or customs that contradict the Sharia. This is supported by the actions of the Companions. You will not be undutiful by requiring one of them to take an oath if justice necessitates it; rather, it is an obligation.