Is it fine if I only apply top coat to the most used (black paint) parts?
To answer this generically, sometimes yes, sometimes no. Adding a clear finish such as lacquer or varnish over paint can offer increased protection, and with some paints this may be needed for the expected use. So a lot depends on the paint chosen.... some paints are more than capable of holding up over extended periods and some high-wear usages without any help.
In your case, based on the further description of the paint you bought from the Comments, there's a decent chance it doesn't need overcoating.
Will oil based polyurethane visibly yellow over black latex paint? How soon will it happen?
Oil-based polyurethane is already slightly yellowish (or more accurately, amber). It just becomes slightly darker and more orange/tan over time with ageing and exposure to light. This darkening takes a few years usually before it's noticeable.
But, you don't really have to worry about this since you basically can't see a yellowish clear finish over a black background.
Two further things to bear in mind however:
The curing period for oil-based poly is long, up to a month. So if you do use it your desk won't be available for normal use, where you don't have to baby it, for at least two weeks and possibly double that, depending on local conditions1.
Matt varnishes don't hold up as well as more glossy variants, in terms of maintaining their original look. The main issue is that the matt surface can become buffed by use, leading to glossier areas especially along edges, but even where you commonly rest the heel of your hand using a mouse or over the edge of a laptop can create a shiny spot.
Water based polyurethane and polycrylic
A few application pointers:
In case the application guidelines for the paint don't make this plain, it is far preferable to paint multiple thin coats than fewer thicker coats. I would aim to apply four thin coats to the top of the desk, and fewer elsewhere2.
In addition to the above the first coat can the thinned quite a bit so that it sinks into the wood more deeply. (IME you can do this safely even if the paint's instructions say not to thin the product.)
I'm personally a big fan of using foam rollers for applying waterbased paints, but you may not like the subtly textured surface this often produces so would prefer to apply the paint conventionally.
1 The rule of thumb is that varnish can be considered cured once you can no longer smell it with your nose right by the surface.
2 On non-critical surfaces once full coverage was achieved I'd generally stop, unless I was feeling particularly conscientious that day :-)