If the PD is a universally applicable moral maxim (which is debatable to say the least), then it applies to any rational being who can comprehend it.
If the PD is binding only on those who have sworn to uphold it, then it would depend on how you construe that oath. Arguably, the oath applies only for the remainder of that particular incarnation. As a wedding vow is “til death do us part”, and not for all eternity, likewise with oaths of allegiance to political entities or ideologies.
Those who believe in reincarnation posit some sort of meta-soul or “real self” that transcends the temporary persona that is attached to a particular incarnation. I think they would say that the “real self” is not necessarily bound by promises made by a persona during a particular incarnation (unless maybe Enlightenment was achieved during that incarnation prior to making the promise).
Edit: on the other hand, if they believe that retribution may be due in one life due to wrongs committed in a previous life, maybe they would consider an oath to be binding across incarnations if the new persona remembers making it. But I don’t think that holds up under scrutiny. I’m sticking with my wedding vow analogy.