Why not "Geschäft" ?
Geschäft is used more in a comercial way. Affäre is more used for an unwanted circumstance or a when you have something going with someone like love affair.
I'm having trouble with the different cases. Why would "Das Verhältnis betrifft MEINEN Eltern." be an incorrect translation for this sentence?
Why not look for a good table of conjugations? Plural article and determiners have "e" in the Accusative case.
Why not Die Sache betrifft meine Eltern?