#implicitly:

TOS doesn't give the slightest implication about any other family member having any meaningful presence or significance in his life at all. We're given no reason in the show to think he has children or other family of his own (it's wildly unlikely for TOS Kirk tbh, for various reasons). We're given no reason to think their parents are in the picture or have been in a long time, certainly none to think either parent was at the ceremony—TOS Kirk is never associated with conventional parent-child nuclear family at all.

It was a friend of their family who helped Kirk apply to the Academy, not their parents. The people mentioned in TOS as guides or inspirations to him, or as people he looks up to, are all unrelated mentors in Starfleet like obvious father-figure Captain Garrovick or public figures like T'Pau. The only "traditional" family mentioned as there for him and whom he ever acknowledges is his older brother, sister-in-law, and nephews—all of whom he's clearly very fond of and has a close relationship with—not parents or children.

We don't know the ages or birth order of Sam and Aurelan's children, but for all three to be of appropriate ages for a significant Starfleet event, Sam is likely to be significantly older than Jim, with the extremely brief glimpse of Shatner!Sam + mustache obscuring how youthful he really looked (the writers had envisioned a 10-year age difference, though this is never mentioned onscreen; it does make the backstory from "The Conscience of the King" work more easily, though, since Sam's survival no longer needs to be explained—he would have already been an adult and off on his own life, if young, and frankly he and Aurelan seem to have done better by a troubled 13-year-old younger brother who had just survived starvation and genocide than Sarek and Amanda ever managed with Spock for... existing).

But the other thing about Sam as Bill Shatner With Mustache is the obvious implication that, while I don't think we're meant to think the brothers are diegetically identical, they do resemble each other very closely apart from age. Jim looks far more like Sam than Sam's one surviving child, Peter, ever will. And Sam is going to be 43 forever—to Jim, to Peter. But Jim lives. Every year his reflection looks a little more like Sam's.

ngl I think quite a bit about this in terms of Viola in Twelfth Night, a character who is already deeply fitting for comparison to James "What is gender, really?" Kirk:

I am all the daughters of my father’s house, And all the brothers, too—and yet I know not.

I my brother know Yet living in my glass. Even such and so In favor was my brother, and he went Still in this fashion, color, ornament, For him I imitate.

Bonus feature:

She [Viola] pined in thought, And with a green and yellow melancholy She sat like Patience on a monument, Smiling at grief. Was not this love indeed?

But there's more!

There is a cut scene from the end of "Operation: Annihilate!" where Kirk takes Peter (in a tiny Starfleet uniform!) to the bridge, and puts him in the captain's chair to ask which offered home he wants to live in: one is with Sam's research partner and his (the partner's) wife on Deneva (Peter's home, despite the horrors, which Sam had called the best place in the galaxy) and the other with his grandmother on Earth (I don't think it was stated whether this was Aurelan's or Sam's mother, though I'm inclined to think Aurelan's for various reasons, including the descriptions of Peter as the last survivor of Sam's family; Kirk describing her as "your grandmother" with no indication that he's talking about his own mother; and Kirk somberly responding "No one ever is" to Peter sadly saying that living with Sam's friends won't be the same as his own parents). Peter chooses to stay on Deneva.

Unsurprisingly, I adore the idea of Kirk not only asking Peter what he wants rather than unilaterally making the decision for him, btw, but sitting him in the captain's chair of the Enterprise—!!!—and lowering himself to nearly Peter's eye level to ask him what he wants. Uncle Jim understands symbolism, okay.

There's also this bit after Peter is beamed back down to Deneva where Scotty says that Peter seems a bright lad who might well end up in Starfleet, and Kirk grimly responds that he hopes not. He explains that he doesn't want Peter to ever have to weigh decisions like destroying a planet to stop the massacre of billions and will follow a different, happier path.

He's fine, they're fine, the remnants of this impossibly suffering family are totally fine.