The king and queen aren’t Chris’s parents, though? We haven’t met them IIRC
but now I want to see Kaito tell Chris’s parents that he seeks an honest relationship with her while having hot sweaty slave sex in front of them
yeah - Chris is a duchess, not a royal princess. We
see her parents after Kaito manages to prevent Chris' mother from dying when they were children, but looking at the first chapters again, we don't actually get a full introduction, nor does Kaito confront them - he simply leaves the kingdom with Chris once the prince chases her off.
But now the King & Queen are looking to stop the "Heroine" of the story from ruining everything alongside their idiot son, and given Chris' family has a lot of political power, they might factor into that confrontation as well as being the "final hurdle" for Kaito to be with Chris in full.