E.P.
The best thing about your post is that your cousin does have a little angst about telling her parents - and she's 24! I'm just curious why she is so freaked out about this.... are her parents paying for college? Are they not close? Terribly strict? Possibly foreign - not the same American values that their daughter has????
She seems like she has been semi-responsible up to this point in her life, so, like everything else she has done, she needs to step up to the plate and tell them face-to-face and (carefully). I say "semi" because, it sounds like her choice of men may be shaky too. Her choice of birth control or lack of it may be concerning too. Yes, her parents may be upset, and rightfully so. She has veered from the path that she was on. Big changes ahead.
Regarding "the dad"...... again, the parents rightfully may be disappointed because, she not ONLY is in college getting her Masters but she's seeing someone who "is a student" and they are "not in the best relationship".... REALLY! Cut the parents some slack. There's always potential to seek counseling but I really hope that her choice to correct this situation doesn't involve marrying someone who she is not totally committed to. Hopefully her parents will offer her some guidance and support once she tells them and the shock wears off.
As a cousin and good friend, it sounds like you will be able to offer her support as well, which is just what she will need in the months ahead. Maybe coming along when she tells them, will serve as a buffer and could be a help. Bring homemade cookies or your twin boys... anything to cause a distraction!
Good luck.