I know that "Jackson jr." means Jackson the junior, but what about Jackson the elder? How do they abbreviate "elder" in American English?
They don't, because the word used is "senior", not "elder". ("Senior" is abbreviated "Sr.")
When referring to relatives, "elder" is most often used for siblings. In this context, its opposite is "younger". Since these are comparative terms rather than labels, there is not generally a need to abbreviate them.