Makes me think that knowing the parents before going exclusive, is mandatory! Could save a lot of grief.
I don't think meeting them is even necessary. I just mention some things about my parents (usually the first or second night I meet a girl)--and they'll tell you everything you need to know.
Red flags to look out for:
-Absent dad (she probably sleeps with a lot of guys)
-A mom whose a user and who "she'll never be like" (she'll be just like her mother)
-A mentally unstable mom (watch out)
-A controlling mother (she'll be controlling)
-Abusive parents (obviously)
-If either mom or dad were habitual cheaters (if mom, she'll probably cheat on you; if dad, she'll probably go cold & avoidant at the first sign of real intimacy)
-A father who is weak, soft, or overly attentive (serial monogamist alert)
-Whether her parents had her at a very young age (high risk pregnancy alert)
None of these things immediately disqualify her, but I'm rarely surprised how things end.