Hmm, I don't get what a persona non grata would have to do with it, tho.
Except if it is linked to ignoring and breaking the given rule / law in a metaphorical sense: No matter what the protagonists or their animal partners would choose, if they don't leash one participant (aside the new non-person category, just plainly related to the rules given there and their animal-like occurrence), they would exclude this one from being welcome at the park and as such a persona or non-persona non grata, whatever it is at the end (animal, human, furry fursona..).
So as to fulfill it in the common sense of "leash your pets" it is necessary to declare a new form of "person" which basically includes fursonas and so on while excluding the "pets".
Or to define the "pet" aspect: one way would be - if it can't understand simple scientific or argumentative arguing based on abstract logic not linked to the thinking animal itself (direct classical conditioning shouldn't play a role here) - then it wouldn't resemble a person.
The "non-person" is one of multiple possibilities to declare it, but somewhat contrary to what many zoophiles would want: granting their animals a set of basic person (in this case human) rights which surpass animal rights. Be it for protection or for a certified relationship status like marriages.
Anyway, off to bed, four days of harshly timed events follow.