Two things that might possibly have held her back: one, her "big break" at Warner Brothers was in a remake of Of Human Bondage, which got her all kinds of bad publicity because she was playing a part that was associated with Bette Davis. Her career may have been sidetracked a bit by the invidious comparisons; if that was her test to see if she could step into Davis's shoes as a WB leading lady, it didn't go well. (I haven't seen the Of Human Bondage remake, though I like Erich Korngold's score and I'm no great fan of Davis or the 1934 film in general.)
And second, Warners in the late '40s had a lot of talented actresses and no real idea of how to use them. Many of my favorite might-have-beens -- actresses who could have and perhaps should have been big stars -- were Warners contract players in the mid-to-late '40s: Janis Paige, Dorothy Malone, Martha Vickers, possibly Alexis Smith. But even though Warners had a need for new female stars after losing Olivia De Havilland, and then Bette Davis, they seemed unable to take advantage of their own pool of talent. Instead they resorted to borrowing leading ladies from elsewhere, or signing second-tier stars. (I like Virginia Mayo, but I can't quite understand why Warner Brothers signed her after she left Goldwyn; they had plenty of actresses under contract who would have been better in some of the parts she got.) The one exception was Doris Day, who of course got her first leading role as a last-minute replacement when they couldn't borrow a bigger star from another studio. Another studio might have been able to make a true star out of Eleanor Parker, but that's speculation; by the time she left Warners and went to MGM, she was typecast as the gorgeous second banana.
I think I'm not the only one who audibly gasped when the Captain chose Julie Andrews over her. Julie was a lovely person who was great with the kids. That's why in the real world the Captain would have married super-interesting Eleanor and taken her on a very long trip around the world, whilst Julie stayed at home, as the nanny, taking care of his children. But it seems to have been Eleanor Parker's fate as an actress to always play the woman who in (incomprehensibly) ditched and sluttified. I like the fact she is too classy to ever be believable in that role.