It took a lot of convincing for Cartman to actually want to take care of a kid. He always thought he’d be bad at it. Kyle eventually convinced him, but it took a year of on and off squabbles before they actually did it. Cartman pretended he just didn’t want a kid because they were annoying, but Kyle eventually pulled the real reason, him not thinking he’d be a good father, out of him.
Cartman is mostly a playmate to his kid. He prefers to be the good cop. Kyle has to play the unfun father role who always reminds Kenneth of responsibilities and learning. But Kyle is also incredibly nurturing.