I can't remember how much of this is implied canon or my own interpretation, but I fully believed that Chiron recognized Percy was a son of Poseidon from the fountain incident with Nancy Bobofit, knew that a child of the Big Three being claimed would be a total shit show, and simply chose not to say a word about it
wait isn't it also implied somewhere that Sally was in contact with Chiron at some point before Percy arrived at CHB? Im dying imagining a convo between them about Percy's godly parentage like-
Chiron: Unfortunately, its fairly common for the gods to not disclose of their godly nature, so a lot of parents don't exactly know who-
Sally: Oh I always knew Percy's father was a god! You see I met him at a beach in Montauk and-
Chiron: You know what Sally 😀 I'm going to have to stop you right there.