The most obvious Mom & Dad would be Janeway & Chekotay on Voyager. They provide a lot of emotional support while remaining figures of ultimate authority, and yet find it difficult to hide their messy emotions from those they command. They're also kind of uncool, as the episode "Bride Of Chaotica" amply proves: Janeway is game enough to play her part in the ridiculous Captain Proton adventure, but when she actually expresses an interest in continuing the kids do what they can to ditch her.
More individually, Janeway makes the decisions and Chekotay sees that they're carried out, doing a gender reverse of the traditional family roles but following a very traditional Commanding Officer / Executive Officer military relationship. Even when there are bumps in the road, for instance Janeway early-on wanting to micromanage Chekotay within his own sphere, they're the kind of bumps that could have fit comfortably into battle-of-the-sexes entertainment of the mid-20th century (but again, reversed).
In this way they're quite similar to Picard & Riker on The Next Generation. Except that Riker does the job in more of an older-brother way. He performs the duties, but spends more time playing jazz and tangling with the locals than counselling the crew. Of course, this is probably because this is the only Trek crew with a Counselor on permanent duty, providing less conditional support compared to the challenging tone of the others. The fact that Riker & Deana Troi are always linked, and often seen working together on various personnel reports & rosters further gives the sense that they are splitting the "Mom" role between them. And when Picard needs to be managed, either because he's avoiding vacation or is having a career crisis in an alternate timeline, it's generally the team of Riker & Troi that get the job done (without technically contradicting his commands, once again like something out of mid-century domestic fiction).
If the Captain / #1 relationships on those two ships are the 24th Century Starfleet family norm (perhaps one being a "normal" family and the other a "blended" one), then DS9, characteristically enough, is more of the "family of choice". Sisko has to be both Mom & Dad to his son, and pretty much to the crew as well, going so far as to feed them...and to nose into their personal lives in a way that Janeway would have suppressed and which would probably not even have occurred to Picard. Kira meanwhile is not at all the kind of XO Riker or Chekotay were. Any time she's doing reports or handling personnel matters it's with the attitude of them being a barely-tolerable tax on her time. She's much more in the mold of a U.S. Vice President: she's there to take over if anything happens to Sisko, and in the meantime to do whatever job she can find for herself, or to be sent off and be Sisko's proxy because he can't be in multiple places at once. In either case, he essentially trusts her to do whatever needs to be done. If Sisko was asked why it was so important for him to keep Kira specifically as his XO, he might reply like Jed Bartlett did about keeping his VP on The West Wing: "I might die."
Of course that leaves a lot of emotional weight on Sisko...even before you add in the extreme diversity of the crew and Sisko's OTHER children: the people of Bajor who see him as the Emissary of the Prophets. So while O'Brien, Worf, and Dax seem pretty comfortable taking problems to him, the rest are largely on their own until their small problems become large ones, contributing to a quite emotionally turbulent crew. Odo sometimes turns to Kira, or gets unsolicited help from Quark, but remains essentially lonely and vulnerable to manipulation by other shapeshifters. Kira generally wants to show a strong face to Sisko, and can't quite swing with Dax's cosmopolitan attitudes, and therefore has to stumble through a series of troubled relationships without much friendly guidance. And Bashir has the trouble of not knowing how troubled he is, which manifests as increasingly inappropriate advances on Dax, controlling relationships over several female patients, a rather infantile sexual relationship with Leeta, and alternatingly halting and reckless explorations with Garak, until he builds a strong brotherly bond with Miles O'Brien.
Bashir brings up the troubling problem with any virtual family: any sex within that family is in some sense incestuous. I find it interesting to reflect that while there are numerous canon pairings on Voyager with either sexual history or tension, almost none of them are between characters who start out as both-Starfleet or both-Maquis. If it's a blended family, the attraction is across the divide: Greg & Marcia Brady rather than Keith & Laurie Partridge. The one exception, Seska & Chekotay, is significantly the one that starts out troubled and quickly spins into destructive.
Similarly incestuous but on-message would be Amanda Rogers' crush on Riker in TNG, which definitely has the vibe of something that should never have been openly expressed. Ditto Barclay. Other TNG crew pairings: Data / Yar, Picard / Dr. Crusher, and various one-episode flings, all carry the message of either "it shouldn't have happened" or "it can't go on" or both.
On DS9, things are somewhat the opposite of Voyager. Starfleet types orbit each other in various combinations (Bashir, Worf & the Daxes) whereas non-fleeters Kira & Odo and Garak & Ziyal hook up with each other. That this is all positively portrayed (though not without drama) shows me that there's a better word for this "family of choice": they're friends, and emotionally at least have an independence that the other crews don't. Although it is pretty telling that, when they do get access to a Counselor, there's plenty to keep her busy (even though she, like the TNG one, has to compete with / accept the assistance of a more charismatic unofficial one).
I'm going to leave out TOS & Enterprise for now. But I of course welcome all input!
Addendum: It now occurs to me that the one exception to the "all sex within the group is incest" rule would be between Mom & Dad, though Star Trek has definite feeling that the commanding officer is so exalted that even the XO can't really meet them on that plane. This question doesn't come up at all in DS9 as there's no spark of that kind, whereas in TNG the closest you get is a moment where Picard is "tempted" by someone in Troi's form who explicitly says he could have something normally forbidden (the Picard/Crusher non-start is somewhat in the same vein, although she's more of a smart aunt to the crew than a mom). Voyager of course is the one with the most elements militating for a hook-up: a more dynamic command relationship, unfamiliarity, remoteness from anyone of higher rank to enforce the rules, plus a situation where an old fashioned royal marriage might help bind the crews together. That level of crew acceptance can be overestimated though, viz the pilots in a similar situation on Battlestar Galactica and how dismissively squicked they are at the thought that their Captain & President might be having sex.