Question:

If a mourner said Kaddish at services and then drops in to another synagogue to help complete the minyan, does he have to say Kaddish again?

Response:

As a rule, if you have already said a certain Kaddish and are present at a second minyan, you need not recite Kaddish again.

Rabbi Ephraim Greenblatt writes that he was once shown a handwritten responsum of Rabbi Moshe Feinstein, where he ruled that an exception to this rule is a case where if a person does not say Kaddish, people would suspect that he is not saying Kaddish for his parent altogether.

Our obligation to our parents is so very strong that we repeat the holy words of Kaddish rather than appear to disrespect them.

As an aside, irrespective of your obligation to your parents, Kaddish must be said at the appropriate places. Thus, if you are present at a minyan where there is no mourner present, you (or someone else whose parents are no longer alive) should recite Kaddish.


Sources:
Rivivos Ephraim II 48:68, Glosses of RaMA to Shulchan Aruch 132:2.