Where does the custom not to say Tachanun on a Yahrtzeit come from?
Taamei Haminhagim (citing Maaseh Yechiel) comments that "when it comes to saying Tachanun, we omit it based on any possible reason, since 'it is better to recite fewer prayers with more concentration [than the reverse].'"
So presumably a yahrtzeit, whether of one's own parents or of a prominent tzaddik, is such a reason.
I used to Daven in a Shul where they almost never said Tachanun as they always found some Yarzeit. One day they had no Yarzeit and they said Tachanun. One of the Alte Yiden in the back Klapped on his table and said "Vus Far Ah Yom Tov Iz Heint?"
The persian jews have such a minhag when there is an avel (c'v's' had a close reltive die within the seven days). We don's say Tachnun when he is in the minyan.