I think the biggest reason why you don’t commonly see the selective incorporation of male DNA is because the machinery to do the selection would be too costly compared to just transitioning to asexual reproduction.
That being said, there’s a wide breadth of parthenogenesis strategies, of which the most relevant are the kleptons, which can sometimes incorporate some of the male DNA.
Individual genes defecting are probably closest to transposons and other selfish genetic elements, and those are in competition with systems that silence them to prevent them from defecting.
Polyploidy probably have greater flexibility for the non-balanced incorporation of DNA, but I’m not familiar enough to comment any more.
From another message:
I guess selective incorporation machinery might not be too costly, but why selectively incorporate when you can just turn to full asexual reproduction, or have both sexual and asexual reproduction? (ie: virgin birth in sharks, some reptiles, etc.)
I guess it’s the difference between having each offspring being an 90⁄10 split of genetics, or having 80% of your population asexually reproduce and 20% sexually reproduce.
From a message on reddit by /u/eniteris:
From another message:
(I am satisfied with this as an explanation)
To put it another way, if a female inserts more of her DNA into an offspring then she loses out on the benefits of sexual reproduction.
It’s not clear this is the case when the sum isn’t 1. (i.e. 1⁄2 + 1⁄2 = 1, versus 1⁄2 + 1 = 3/2*)
*I’m guessing that’s the breakdown.