The absolute worst father in Dragon Ball Z has got to be King Cold, a man willing to replace his favorite son with his murderer in a heartbeat. Cooler acknowledges in the movie Cooler's Revenge that King Cold favored Frieza over him, indicating that somehow this treatment of his son was actually positive compared to how Cold treated Cooler.
This begs the question: why? Despite Frieza being by all accounts weaker than Cooler -- who, after all, achieved a fifth form long before Frieza uncovered his Golden Form -- King Cold preferred his younger son. What qualities did Cooler possess that let him fall out of Cold's favor?
Did King Cold Really Spoil Frieza Over Cooler?
Due to Cooler's Revenge's complicated place in Dragon Ball Z canon, it's hard to determine how much of what Cooler said should be taken into consideration. The film should take place immediately after the Frieza Saga but before King Cold's arrival on Earth, due in part to Gohan's hairstyle and Cooler believing that Goku killed Frieza rather than Frieza being rebuilt by his father and slain by Trunks. There are obvious continuity issues with that placement (Goku wasn't on Earth at the time), but this establishes that Cold lost both of his sons in quick succession.
However, King Cold goes out of his way to rebuild Frieza while allowing Cooler to return as Metal Cooler on his own through some Lovecraftian machines. Cold also helps Frieza get revenge on Earth, while offering Cooler little to no help -- though, bizarrely, Cooler came far closer to killing Goku and friends than his brother and father did. Yet when King Cold retired from his planet trade organization -- shown in Dragon Ball Super: Broly -- he announced his younger son Frieza as his successor and gave him the keys, rather than Cooler.
The Accomplishments of Frieza vs. Cooler
One possible reason why King Cold might've preferred Frieza is that he seemed to accomplish more than Cooler, even though the older sibling proves stronger during their lifetimes. Dragon Ball Super shows that Frieza never trained, while Cooler actively sought a new and more powerful form, presumably through training. However, while Cooler has an army, it's never shown to be as massive or as influential as Frieza's.
In the years that follow, Frieza's forces prove fiercely loyal to him. When he dies, his troops come to revive him -- more than once, as Super proves. With Cooler, his Armored Squadron seem to be the only people loyal to him. Frieza destroyed Planet Vegeta after using the Saiyans as his personal army for generations. On the other hand, Cooler just watches and criticizes Frieza, only to do nothing.
However, Cooler's accomplishments are entirely based on his own efforts. Frieza was given power by his father. All of his accomplishments are a result of him inheriting that power, rather than earning it like Cooler. Frieza, if anything, weakened King Cold's empire by killing almost all of the Saiyans out of sheer paranoia.
King Cold Spoiled Frieza Because He Was Younger
There is a stereotype with parents. They are harder and stricter with their elder children, but the younger ones are spoiled or given a looser leash. The same dynamic is shown between Gohan and Goten. While Gohan is pressured to pursue academics, Chi-Chi lets Goten do largely whatever he wants. While that relationship is a constructive one, it's likely that the same dynamic held true for Frieza and Cooler with their father. Cooler was held to a higher standard while Frieza was given more free reign.
It seems as if King Cold prioritized giving Frieza everything while Cooler, held to a higher standard, had to accomplish everything on his own. While it's uncertain how much he actually accomplished, it's clear that Cooler had little to no support from his father doing so. Perhaps Cold hoped that Cooler could create his own empire, only for him to fall at the hands of Goku and friends.