Given that Spock served with his crewmates for 3 years before mentioning that his father was the Vulcan ambassador...
He actually mentioned it a bit sooner than "Journey to Babel." In "This Side of Paradise" (also written by D.C. Fontana), Spock mentions his parents' occupations when Kirk is taunting him, although he's still referring to them in the past tense:
KIRK: What can you expect from a simpering, devil-eared freak whose father was a computer and his mother an encyclopedia?
SPOCK: My mother was a teacher. My father an ambassador.
KIRK: Your father was a computer, like his son. An ambassador from a planet of traitors. A Vulcan never lived who had an ounce of integrity.
So Kirk maybe shouldn't have been
quite so shocked when Spock revealed that Sarek the Vulcan Ambassador was his father the next season.
It's interesting how Spock's references to his parents get gradually less vague as TOS goes on. In WNMHGB, he says that "one of my ancestors married a human female." In "The Corbomite Maneuver," we get this exchange:
SPOCK: I regret not having learned more about this Balok. In some manner, he was reminiscent of my father.
SCOTT: Then may heaven have helped your mother.
SPOCK: Quite the contrary. She considered herself a very fortunate Earth woman.
Notice that both of his parents are referred to in the past tense.
"Mudd's Women" had Harry Mudd recognize Spock as "half-Vulcanian" somehow, presumably because full Vulcanians looked noticeably different than Spock (perhaps they were closer in appearance to the fake Balok puppet?). This exchange of course didn't make as much sense once it was shown that full Vulcans look more or less the same as Mr. Spock.
The next episode, "The Enemy Within," reinforced that Spock was half-human and half-Vulcan:
SPOCK: Being split in two halves is no theory with me, Doctor. I have a human half, you see, as well as an alien half, submerged, constantly at war with each other. Personal experience, Doctor. I survive it because my intelligence wins over both, makes them live together. Your intelligence would enable you to survive as well.
In "The Naked Time," Spock says:
SPOCK: My mother. I could never tell her I loved her.
KIRK: We've got four minutes, maybe five.
SPOCK: An Earth woman, living on a planet where love, emotion, is bad taste.
KIRK: We've got to risk a full-power start. The engines were shut off. No time to regenerate them. Do you hear me? We've got to risk a full-power start!
SPOCK: I respected my father, our customs. I was ashamed of my Earth blood.
So they nailed down that his father was Vulcan and that his mother was human, but they were still referred to in the past tense, presumably dead.
In "The Squire of Gothos," Spock refers to his father in the present tense for the first time:
SPOCK: I am Spock.
TRELANE: Surely not an officer. He isn't quite human, is he?
SPOCK: My father is from the planet Vulcan.
TRELANE: And are its natives predatory?
SPOCK: Not generally. But there have been exceptions.
Then the parents' occupations are nailed down in "This Side of Paradise," late in the first season.
AFAIK, "Amok Time" didn't make any reference to Spock's parents, outside of the mention of them arranging his marriage to T'Pring as a child. The fact that we didn't see them in the episode is perhaps another indicator that the TOS writing staff regarded them as dead at this point in time.
And at the beginning of "Journey to Babel," Kirk apparently knows that Spock's parents are alive:
SAREK: I'd prefer another guide, Captain.
KIRK: As you wish, Ambassador. Mister Spock, we'll leave orbit in two hours. Would you care to beam down and visit your parents?
SPOCK: Captain, Ambassador Sarek and his wife are my parents.
I guess by this point, Kirk had just accepted that Spock was an extremely private person, and didn't pry into his personal history very much.
