I'm the first in my family to go Greek.
My mother was "kind of" supportive - she didn't really know what it was all about, since she went to school overseas, but basically her attitude was "if you want to join a sorority, go ahead, but you have to pay for it". Which is reasonable.
My dad was vehemently opposed. I gather that he had a negative experience with the Greek system on his campus. (Maybe he rushed and didn't get a bid - I don't know.) Then again, my dad is vehemently opposed to just about anything I do - too bad for him that I'm an adult and can do as I please