Yes, a reasonable interpretation, from the perspective of today. However, be aware that the passage in question may have been written based on the premise that the Enterprise was currently in the fifth year of its ongoing mission. I'm not exactly sure what the phrase "five year mission" in the opening monologue was originally intended or understood to mean, nor at what point(s) any such intent or understanding might have changed. Maybe someone like @Harvey or @Maurice could provide greater insight into that? Anyway, as for what was ultimately onscreen, Kirk said in "Amok Time" (TOS) that Spock hadn't asked for any sort of leave "in all the years" he'd known him, and had in fact refused them. Nine episodes later (by either production or original airdate order) in "Journey To Babel" (TOS), Amanda says Spock hadn't been to see her and Sarek in four years. Earlier, in "Where No Man Has Gone Before" (TOS), Dehner said Spock had worked next to Gary Mitchell "for years" at that point, while also seeming to imply that Mitchell had served on Kirk's "first command"—which, as the TMoST passage clarifies, was not intended to refer to the Enterprise herself—though ultimately only saying that Kirk "asked for" Mitchell on that unspecified assignment. These references would indeed seem to comport with a scenario where Kirk and Mitchell came aboard the Enterprise from their previous assignment together, at that point both meeting Spock for the first time, with Kirk meeting Pike for the only time per "The Menagerie" (TOS), and this would all be at least two years before WNMHGB, but no more than four years before "Babel"... Of course, depending on the understanding and storytelling desires of a given writer, there might be wiggle room for other interpretations. For instance, speaking strictly canonically and discounting production intent never made explicit, Kirk's first command could indeed have been the Enterprise. Or, if not, Kirk could indeed have asked for Mitchell aboard it but not received him, because Mitchell had already been assigned to the Enterprise, where he was already working with Spock, before Kirk ever came aboard. Kirk could even have met Spock way back at the Academy, despite not having met Pike until later, and simply been unaware of any leaves Spock might or might not have taken before he became his commanding officer. And perhaps no sort of leave was required in order for Spock to visit his parents that previous time, due to whatever circumstance. Heck, Amanda could have been speaking in terms of Vulcan years, for that matter! "There always are...possibilities." -MMoM
The only thing the "five year mission" means is they wanted to get 5 seasons of episodes for the ideal sydication package.