The entirety of Karl's dad's on-screen time can be squeezed into one brief YouTube video, but his effect on his son's life colors the entire movie. Dad's parenting style is summed up by this quote:
DAD: I ain't got no boy. […] You ain't no kin to me.
We have to wonder if Dad understands how complex his statement is. This man had two sons, and now he says he has no boy. He asked Karl to kill the other baby, born prematurely, and now he has disavowed the existence of both his offspring. Dad then put his surviving son in a shack and left him there.
On top of that, this guy has a house decorated with Christian artifacts and assumes a holier-than-thou, sanctimonious attitude, even though he is a complete failure of a father. The words "heartless" and "cruel" don't even begin to cover this man's actions.
After having his existence disavowed in front of him, Karl gives a speech to his dad, maybe his longest in the film besides the opening monologue. Here's a snippet of it:
KARL: I studied on killing you. I studied about it quite a bit. But I don't reckon there's no need for it if all you're gonna do is sit there in that chair. You'll be dead soon enough. And the world'll be shut of you.
This speech shows us that Karl has forgiven his father, in his way, but it also serves as a bit of misdirection. It makes us think that Karl has realized that killing isn't the answer. But it's shortly after this episode that Karl up and kills Doyle. Karl knows his dad is at death's door, but Doyle is still a few hundred miles from his final destination, so Karl takes it upon himself to speed up Doyle's journey to meet his maker.
Back to Karl's dad. There is a weird, sentimental tenderness to Karl's relationship with his father. His pops may be an awful man, but he's still Karl's dad. And he did instill in his son a lifelong love of biscuits and mustard, so there's that.
We're not sure the exact motive for Karl's late visit to his father, but it seems that he goes there seeking acceptance, which he has never had for his entire life. Maybe Karl thinks that his dad, as he is dying, will repent for his past sins.
Yeah, not happening. The miserable old jerk is still a miserable old jerk, while Karl steps up as the more mature individual by showing his father mercy and deciding to spare the guy's life—a mercy his younger brother never had.
If the mental hospital allows it, Karl should send his dear old dad a Father's Day card. Maybe something heartfelt, like Dear Dad, Hope You Are Dead Now, Love Karl.