It's not モロコネ, but two words, モロ & コネ.
モロ is slang form of "obviously", and コネ is an elliptical word of "connection" from English!
And, the エゴ is also from English, "egoism", so "親のエゴってやつ?" is like "It's from her parents' egoism, isn't it?"
I guess.
(By the way, do you like doctors...