This is a fan theoy based on the manga. I have no idea where to put it so I'll put it here.
My theory is that Kruger told Zeke to tell on his parents so that Grisha Yeager could inherent his Attack Titan as close to the walls as possible.
Kruger was close to death and he had to give away his powers. So he was looking for someone that could bring back the Founding titan and restore the Eldian empire. He had noticed Grisha's drive and intelligence and chose him for the job. Now how would Kruger give his powers? The only way is to turn Grisha into a mindless titan and then be consumed. The only place that Kruger had access to the shots of spinal fluid is at the wall at the shore of Paradis Island. But one problem remains, how is Grisha going to get there? Kruger must have seen what a bright kid Zeke was and he knew that Zeke wanted to free his people. So he told him to tell on his parents. I imagine that Zeke must have had a thousand questions and I think that Kruger answered them, and he also told him his entire future plan which must have convinced Zeke to betray his parents and is also the reason why Zeke betrayed Marley.
The similarity between these two characters are huge, they both work inside the Marleyan military, they both have goals to free their people, both have titan powers and both are willing to sacrifice everything for their goal. I don't think it's a stretch to say they share the same plans.
I find it difficult to believe that Zeke would hatch his plan to turn in his parents all on his own given that he was about seven at the time. Most seven year olds have trouble plotting how to not get in trouble at school, let alone how to become a solo agent in hostile territory.
Planning with Zeke doesn't feel like something that Kruger would have done, because he's been incredibly indirect the entire time to avoid drawing suspicion to himself, but it's plausible.