The issue of Kirk not knowing about Spock's parents is a thorny one. Remember that he doesn't know much about Spock's family at
any timepoint: he has no idea about the bethrothal to T'Pring - or about the existence of Sybok, decades later.
Much of this must be attributed to Spock's secretive nature. Remember that he essentially eloped to Starfleet against his father's wishes. His rare enrolling might have been treated with such glee that he was granted the Foreign Legion treatment: no questions asked. He certainly didn't confide in his crewmates - he kept talking about his parents in the past tense until confronted with the fact that they were still alive and making themselves known to his crewmates. He could have developed an extensive net of lies about his shameful family background, all under the classic Vulcan pretense of "we never lie".
If Spock indeed joined under a Foreign Legion clause, it could be that Kirk would be handed a very incomplete personnel sheet. He couldn't complain much, either: he would have inherited Spock from Pike, not handpicked the officer himself (like other Starfleet captains have been shown doing, in contrast with real-world practices). Heck, Starfleet probably forced the experienced Spock on the young Kirk, mindful of their precious starship...
On a related note, Kirk didn't know that Chekov was an only child. Whether he would have been aware of McCoy's daughter and failed marriage in the original version of "Way to Eden", I don't know... The existence of Sulu's daughter isn't really a surprise for him in ST:GEN, but does accentuate the fact that he doesn't closely follow the family lives of his officers.
Assorted nitpicks:
We already know that Kirk is aware Spock’s mother is human and that his father is a Vulcan ambassador because Kirk taunted him with it in This Side of Paradise.
As said, Spock always implied they were dead, including in this episode - and Kirk speaks of them in the past tense here, too.
Also, Kirk does
not know these things in "This Side of Paradise". He
learns them from Spock during that scene. Arguably, he should retain the knowledge in "Amok Time" and "Journey to Babel" (which both have later stardates) - but for all he knew, Spock's dad as of "Journey to Babel" was a former Ambassador, now decomposing in some rocky grave on Vulcan.
Unless I am way off base, it is not standard Vulcan procedure for an ambassador to marry a human woman and have a hybrid child. It’s safe to say Sarek is the only one who did this at this point.
All the more reason to hide this shameful incident of zoophilia from the interstellar public...
Although we could argue that one of the prerequisites for becoming an interstellar Ambassador is to marry an alien.
Yet, he still breaks protocol and – in front of a dozen guards and the newly arrived ambassador – loudly asks Spock if he wants to go down and visit mommy and daddy. Forget that it’s Spock, how is asking this proper behavior toward ANYONE in this situation?
Umm, it wouldn't be a breach of protocol at all in case of a human underling - it would be courtesy when the CO offered personal leave to his trusted officer, and all the more so when the CO made his trust on the officer so openly known.
And I beg to differ - Kirk has never shown any particular understanding towards Spock's need for personal secrecy. He brings out the pliers in "Amok Time" when McCoy is ready to stay with the velvet gloves. He confronts Spock's Vulcan ways basically as often as McCoy does, although
he doesn't have to stoop to arguments because his word is final. And yes, he likes to make an ass of Spock on the bridge at the end of virtually every episode...
Timo Saloniemi