Learn Hardcore Icelandic: Foreldrar mínir láta mig fara snemma að sofa á virkum dögum. - My parents make me go to bed early on weekdays.

foreldrar mínir = my parents (normal, neutral way to say it)
mínir foreldrar is possible, but it sounds marked and usually adds emphasis, like my parents (as opposed to someone else’s parents).

So foreldrar mínir is the standard, unmarked word order.

Why is the form mínir used here and not minn, mín, or mitt?

The possessive pronoun must agree with the noun in gender, number, and case:

foreldrar = parents
- gender: masculine
- number: plural
- case: nominative (subject of the sentence)

The nominative plural masculine form of minn is mínir, so we get:

foreldrar mínir = my parents

Other forms for comparison (nominative only):

masculine: minn (sg), mínir (pl)
feminine: mín (sg), mínar (pl)
neuter: mitt (sg), mín (pl)

What exactly is the structure of láta mig fara? Why do we use láta plus another verb?

Láta is a verb that often works as a causative: to make / let / have (someone) do something.

The pattern is:

[subject] + láta + [object in accusative] + [infinitive verb]

In your sentence:

Foreldrar mínir – subject (my parents)
láta – finite verb (make / let)
mig – object, accusative of ég (me)
fara – infinitive (to go)

So Foreldrar mínir láta mig fara… literally =
My parents make me go…

No extra word like English to (as in make me to go) is used; Icelandic just uses the bare infinitive fara.

Why is it mig and not mér after láta?

Láta takes its object in the accusative case. The pronoun ég declines like this:

nominative: ég (I)
accusative: mig (me – as a direct object)
dative: mér
genitive: mín

Since I am the thing being caused to do something, I’m a direct object, so the accusative is required:

Foreldrar mínir láta mig fara… = My parents make me go…

Using mér (dative) here would be ungrammatical.

What does fara snemma að sofa literally mean, and why not just sofa snemma?

Literally:

fara – to go
snemma – early
að sofa – to sleep

So fara snemma að sofa is literally to go early to sleep, which corresponds idiomatically to English to go to bed early.

If you said sofa snemma, that would mean to sleep early (focusing on the time of the actual sleeping), not on the act of going to bed. In Icelandic, for go to bed, people usually say:

fara að sofa – go to sleep / go to bed
fara í rúmið – go into bed

So láta mig fara snemma að sofa is the natural way to express make me go to bed early.

Why is the adverb snemma placed before að sofa and not somewhere else?

The phrase fara að sofa behaves like a unit: go-to-sleep / go-to-bed. The adverb snemma usually comes close to the verb it modifies and is often placed before the infinitive phrase in such constructions:

fara snemma að sofa – go early to sleep

You can technically say fara að sofa snemma, and it would still be understood as go to sleep early, but fara snemma að sofa is very natural and keeps fara and að sofa tightly linked as a single idea.

Why is it á virkum dögum and not some other case? Why dative here?

The preposition á can take either accusative or dative, depending on the meaning:

Accusative: movement onto / into something
Dative: location or time (on, in, at in a static sense)

For time expressions like on Mondays, on weekends, on weekdays, á takes the dative:

á mánudögum – on Mondays
á helgum – on weekends
á virkum dögum – on weekdays

So dögum is dative plural, and the adjective virkur agrees and becomes virkum (dative plural masculine).

Why is the adjective virkum in that form, and how does it agree with dögum?

The noun:

dagur – day (masculine)
- dative plural: dögum

The adjective:

virkur – active; in this phrase virkur dagur means a weekday / working day
- dative plural masculine (strong declension): virkum

Adjectives must agree with the noun they describe in gender, number, and case:

masculine + plural + dative → virkum dögum

So á virkum dögum = on (during) working days / weekdays.

Could I also say virka daga instead of á virkum dögum? Is there a difference?

Yes, you can; both are idiomatic:

á virkum dögum – literally on working days (dative with á)
virka daga – on weekdays / on working days (accusative plural without a preposition, used adverbially)

The meaning is basically the same (on weekdays).
Á virkum dögum may feel slightly more explicit and formal; virka daga is a bit shorter and very common in everyday speech.

What is the basic word order in this sentence, and where is the main verb?

The sentence:

Foreldrar mínir láta mig fara snemma að sofa á virkum dögum.

Breakdown:

Foreldrar mínir – subject (my parents)
láta – finite (conjugated) verb in 3rd person plural present
mig – object (me, accusative)
fara snemma að sofa – infinitive phrase (to go to bed early)
á virkum dögum – adverbial time phrase (on weekdays)

Icelandic main clauses usually have the finite verb in second position (V2 word order). Here:

Foreldrar mínir (subject phrase)
láta (finite verb)

Everything else follows after.

Is láta here present tense? It looks the same as the infinitive. How can I tell?

Yes, láta here is present tense, 3rd person plural. The verb láta is a bit special because its infinitive and its 3rd person plural present form are the same in spelling:

infinitive: að láta – to make / let
3rd pers. pl. present: þeir láta – they make / they let

In your sentence, the subject foreldrar mínir is plural, so the verb must be plural present:

Foreldrar mínir láta mig… – My parents make me…

You know it’s finite (not infinitive) because:

there is no að before it
it comes in the typical finite-verb second position
it agrees in number with foreldrar mínir.