I'm not one of those fans who see parallels and automatically think dark, judgmental thoughts. I see parallels as significant resonance in storytelling, affirming what has come before by addressing it and saying something new, which is the basis of all good storytelling. That being said, Star Trek Beyond brings plots from several key episodes you might want to keep in mind:
"The Tholian Web" (original series)
As an episode straight out of the bedrock franchise material from the '60s TV show, "Tholian Web" probably needs the least amount of introduction. Known as much for the introduction of the Tholians (not actually seen until the Enterprise two-part "In a Mirror, Darkly"), it features, like Beyond, an lost Starfleet ship, the Defiant (Deep Space Nine later featured a warship called the Defiant, too), which Kirk finds himself aboard, separated from the rest of the crew (leaving, as in Beyond, Spock and McCoy to speculate about what comes next).
"Power Play" (The Next Generation)
Another lost Starfleet ship, the Essex, comes from the same general era as Beyond's Franklin, and Enterprise's namesake predecessor to the classic franchise line of flagships. As in Beyond, Picard is confronted with a baffling situation focused on a Starfleet captain who isn't what he appears to be. Both "Tholian Web" and "Power Play," as well as by extension Beyond, are a nod to the classic maritime problem of the potential for ships to be lost at sea, which in the case of "Power Play" and Beyond are indicative of the early problems Starfleet encountered in that regard.
"The Swarm" (Voyager)
Aside from similarities in the name of the episode and Krall's army of killer drone ships (hey, has anyone else mentioned that parallel with our modern warfare, the drones?), the way Kirk defeats Krall's drone fleet in Beyond is similar to how Janeway overcomes her enemies in this episode, too. (There's a nice subplot involving the holographic Doctor in desperate need of repair, and getting help from a facsimile of his creator, Lewis Zimmerman, which might be considered in metaphor as a version of Beyond's commentary on the state of Starfleet, then and now.)
Taken as a whole, these episodes don't quite add up to Star Trek Beyond, just as the movie isn't merely a conglomeration of these, or any other, episodes. Always be wary of commentators who gloss over creative decisions like that.