The rest api doesn't provide any way to find other builds that were triggered by a parent build. It seems the best that can be done currently is scraping the console output.

Note that you can go the other direction – figure out which builds triggered any given build – by using the "causes" field in the rest api. But you can't go the other direction.

See http://stackoverflow.com/questions/33104913/getting-subproject-builds-via-jenkins-api