Learn Hardcore Russian: Родители не позволяют сыну долго сидеть за компьютером вечером. - The parents do not allow their son to sit at the computer for a long time in the evening.

In this sentence we’re talking about a general rule / habit: the parents generally do not allow the son to sit at the computer for a long time in the evenings. For repeated or habitual actions, Russian uses the imperfective aspect → не позволяют.

Не позволят would sound like they will not allow (on a particular occasion / in the future), which is not the idea here.

Why is it сидеть за компьютером, not сидеть на компьютере or something else?

In Russian, being “at” a desk / table / computer / piano is usually expressed with за + instrumental:

сидеть за столом — to sit at the table
работать за компьютером — to work at the computer
сидеть за компьютером — to sit at the computer (often meaning using the computer)

So:

за компьютером = at the computer (instrumental)
на компьютере would literally mean on top of the computer, which is not what we mean.

Why is it долго сидеть, not сидеть долго? Does word order change the meaning?

Both долго сидеть and сидеть долго are grammatically correct and very close in meaning.

долго сидеть за компьютером sounds slightly more neutral and natural here, with долго directly modifying сидеть.
сидеть долго за компьютером is also possible, and may place a bit more rhythmic or light emphasis on долго, but not a big change in meaning.

So the word order is flexible; долго сидеть за компьютером is just the more typical phrasing in this context.

What part of speech is долго here and why not долгий or долгое?

Долго is an adverb, meaning “for a long time” or “long” (in duration). It describes how the son sits (the duration of the action), so an adverb is needed:

долго сидеть — to sit for a long time

Долгий / долгое are adjectives and describe nouns, not verbs. For example:

долгое время — (for) a long time
долгий разговор — a long conversation

Here we are not describing a thing, but the duration of the action сидеть, so we use долго.

Why is it вечером instead of в вечер or по вечерам?

Вечером is an adverbial form meaning “in the evening” (when?). Russian often uses the instrumental form of time-of-day nouns as adverbs of time:

утром — in the morning
днём — during the day
вечером — in the evening
ночью — at night

We normally say:

вечером — in the evening (on this evening / in the evenings generally, depending on context)
по вечерам — in the evenings (more explicitly “on evenings, usually/regularly”)

In this sentence, вечером can already express a general habitual time, so по вечерам is possible but not necessary:

Родители не позволяют сыну долго сидеть за компьютером по вечерам.
Slightly stronger sense of “in the evenings (as a regular thing)”, but the meaning is very close.

В вечер is not used in this sense; we say вечером, not в вечер, for “in the evening.”

How does the structure позволять + dative + infinitive work? Is it always like “allow someone to do something”?

Yes, the construction is:

позволять / позволить + кому? (dative) + инфинитив

It corresponds closely to “to allow / permit someone to do something”:

Учитель позволяет детям говорить по-английски.
The teacher allows the children to speak English.
Родители не позволяют сыну долго сидеть за компьютером.
The parents do not allow the son to sit at the computer for a long time.

So you can usually translate it as “allow + (dative) person + to + verb” in English.

Could we say не разрешают сыну instead of не позволяют сыну? Is there a difference?

Yes, you can say:

Родители не разрешают сыну долго сидеть за компьютером вечером.

Разрешать / разрешить also means “to allow / to permit.”

Differences in nuance:

позволять — slightly more general, often about what is acceptable / allowed, sometimes with a nuance of giving someone freedom.
разрешать — more like giving or not giving official permission (rules, authority, etc.).

In this context, both are natural and the difference is small.

Does сидеть за компьютером necessarily mean physically sitting, or can it just mean “using the computer”?

Literally, сидеть means “to sit”, and сидеть за компьютером is literally “to sit at the computer”.

In everyday speech, though, сидеть за компьютером often implies “to be spending time at / on the computer”, i.e., using it, browsing, playing games, etc., not just the physical act of sitting.

So the phrase is usually understood as “spend time on the computer”, which is why the English translation is often more flexible.

Can the word order be changed, for example: Родители сыну не позволяют вечером долго сидеть за компьютером?

Yes, Russian word order is relatively flexible, and your variant is grammatically correct:

Родители сыну не позволяют вечером долго сидеть за компьютером.

Different word orders can change emphasis:

Родители не позволяют сыну долго сидеть за компьютером вечером.
Neutral; focus on the general fact that parents don’t allow this.
Родители сыну не позволяют вечером долго сидеть за компьютером.
Slightly more emphasis on сыну (to the son specifically) or on не позволяют as a restriction directed at him.

All these versions keep the same basic meaning.