Specifically, what is the relationship between PersistentSet and ThirdPartyPersistentSet?
The solid diamond <|>-----> is Composition
("has a") where the "parts" are destroyed when the "whole" is. In the image below, if you destroy a Car, you destroy the Carburetor.
The empty diamond < >-----> is Aggregation
("has a") where the "parts" might are not destroyed when the "whole" is. In the image below, if you destroy a pond, you don't necessarily destroy the ducks (they move to a different pond if they are smart).

(source: wikimedia.org)
What is the relationship between PersistentObject and ThirdPartyPersistentSet?
This is a dependency relationship. See my answer here for further information.
So when does a dependency relationship change to an association relationship when using parameter passing?
If you store the parameter locally, then it changes from a dependency relationship, to an association relationship. If you only use the parameter locally, then it stays a dependency.
C# Code example:
// Association
public class ThirdPartyPersistentSet
{
private PersistentObject _object;
public ThirdPartyPersistentSet(PersistentObject obj)
{
_object = obj; // Store it to a local variable.
// Now ThirdPartyPersistentSet 'knows' about
// the PersistentObject.
}
}
// Dependency
public class ThirdPartyPersistentSet
{
public ThirdPartyPersistentSet(PersistentObject obj)
{
obj.GetSomething(); // Do something with obj,
// but do not store it to a local variable.
// You only 'use' it and ThirdPartyPersistentSet
// does not 'know' about it.
}
}