Minecraft wiki entry for JSON_format, associated text:  conditions:
 child: The child that results from the breeding. May also be a list of predicates that must pass in order for the trigger to activate.
All possible conditions for entities[show]
 parent: The parent. May also be a list of predicates that must pass in order for the trigger to activate.
All possible conditions for entities[show]
 partner: The partner. (The entity the parent was bred with) May also be a list of predicates that must pass in order for the trigger to activate.
All possible conditions for entities[show]
 player: The player that would get the advancement. May also be a list of predicates that must pass in order for the trigger to activate.
All possible conditions for entities[show]