Are multiple scenes filmed at the same time?

Someone might be able to find specific info for Star Trek, but there is no general rule in the industry.

Scenes are rarely shot in order. They are shot according to actor and set/location availability. If they have to build some special set to be an alien ship's bridge for just one episode, they'll shoot everything they can on it so they can break it down again in time to re-use the space for the next special build.
Not really applicable to Star Trek or anything shot entirely indoors but relevant to many productions is crew availability and rest time. If they need several night scenes, then in order to maintain the crew's 'right to rest period' they will film maybe a whole week of nights, everything they need for a whole block, then next week go back to standard days. You can't film a night shoot immediately followed by a day shoot - everybody must get a minimum time between wrap and call.

The rest depends on budget, project duration and also whether they can film scenes simultaneously that don't require the same director. Also dependant on budget is how long they are given to shoot each scene. Some rapid-fire shows can hammer through 12 scenes a day; most will be looking at maybe 5, max. High budget shows down to one or perhaps two. [Massive budget might spend two weeks on a single scene, but that's rare for TV].

Simple establishing shots or scenes with light dialog might be given to a second unit, but the main characterisation, look and feel of a show is usually in the hands of one director per block. A block could be anywhere from a single episode to three or four, or far more rarely these days, the entire season.

Soaps and long seasons - 26 episode shows - may have more flexibility in this, with two or more units concurrently filming separate blocks, with different directors. These tend to be booked around availability of the main cast, so it's not impossible Riker, Troi, Crusher or Data could potentially be shooting one episode in the morning on one stage, then another in the afternoon, on a different stage. Production will try to keep actors employed in such a way as to keep just one plot at a time in their 'head space' but it's not always possible.

A competitor of sorts to Star Trek, Babylon 5, did some amount of this. There were always significant cost pressures, and so one of the things they did was plot out when different sets would be needed, and actors, and so on, so that they could use actors and sets in parallel. They even did this across episodes, so an actor might be filming one episode one day, another episode another day, and then back to the first episode on day 3.

I'm going to have to see if I can find references. I know jms wrote about this in his usenet posts.

In the television world, Star Trek is what is known as a "single camera" show. There is literally just one camera, and there will only be one scene filmed at a time. But ironically, even if it were a "multi-camera" show, there would still only be one scene shot at a time, so industry terminology might not be very helpful here.

The primary reason why there would be only one scene shot at a time is because there is only one director. One director cannot monitor or manage multiple scenes simultaneously. That works both ways. The director cannot know what he is getting, and the actors (and crew) cannot know if they are giving him what he wants, if he is not there to see it.

Generally speaking, the only time there will be multiple crews shooting scenes independently for the same production is when there is a "second unit" involved. Second unit filming is typically confined to establishing shots that don't involve actors. A car driving down the highway, a plane taking off, people walking into a bank.

There are movies made with two directors working simultaneously. The Russo brothers and the Coen brothers are the best examples. Because of the complexity of Marvel films, the Russo brothers could have operated independently at times, but I don't know if they actually did. I doubt the Coen brothers ever did. There is a saying in the business: "Movies with multiple directors look like movies with multiple directors," but that applies to movies where the director was replaced during filming.

As a teen I was an extra on an episode of a popular fantasy show shot in New Zealand. The process was quite interesting, especially as a fan of the show.

This show used a second unit to film distant, static and filler shots using body doubles, often at the same time the main unit was shooting close-ups and walking shots on the main cast. I watched a bit of both while waiting to be called for the few scenes I was loitering in, and the main unit was the only one to shoot close-up and dialog. Even if it was just faces in a crowd reacting to dialog - usually read by an AD rather than the actor - it was in front of the main unit. We wrapped in two days for the episode, including reshoots.

To be fair the episode had two leads, about five guests and about that many feature extras, and the story didn't call for much in the way of split action. There was one scene pair that was filmed back-to-back on opposite sides of a papier mâché cliff face but that was as close as we got. Second unit was off at the time filming the back of a couple of body doubles riding away.

I had a lot of time to talk to the other extras and the body doubles, and this appeared to be the norm at the time for production companies in the area. Any time you see a main character or an important bit of dialog, the Director was sitting on set making sure it fits his "vision."