Báðir foreldrar mínir vinna í borginni.

Because báðir has to agree with foreldrar in gender, number, and case.

Here, foreldrar is:

• plural
• nominative
• masculine in the plural

So both must also be masculine plural nominative: báðir.

A very common source of confusion is bæði, because bæði often appears in expressions like:

• bæði ... og ... = both ... and ...

But in this sentence, both is directly describing parents, so it behaves like an adjective and must agree with the noun:

• Báðir foreldrar mínir ... = Both my parents ...

This is one of those Icelandic patterns that feels strange at first.

The singular word is:

• foreldri = parent (neuter)

But the plural is:

• foreldrar = parents

In the plural, the word behaves as a masculine plural noun, so words that go with it also appear in masculine plural forms:

• báðir
• mínir

So even though the singular is neuter, the plural form you see here takes masculine agreement.

Both orders are possible, but they do not feel exactly the same.

• foreldrar mínir = the normal, neutral way to say my parents
• mínir foreldrar = more emphatic, like my parents as opposed to someone else’s

So:

• Báðir foreldrar mínir vinna í borginni. sounds natural and neutral.
• Báðir mínir foreldrar ... would sound more marked or emphatic.

Icelandic often places possessives after the noun in everyday speech and writing.

Because it has to agree with foreldrar.

Foreldrar is plural, so my must also be plural:

• minn = my (masculine singular)
• mín = my (feminine singular)
• mitt = my (neuter singular)
• mínir = my (masculine plural)

Since foreldrar is masculine plural nominative, the correct form is mínir.

Yes, word-for-word it is roughly parents my, but in Icelandic that is a normal way to express possession.

English usually requires:

• my parents

Icelandic can often say:

• foreldrar mínir

This is not unusual or poetic; it is standard grammar.

Vinna here is the present tense plural form of the verb að vinna.

The verb að vinna can mean:

• to work
• in other contexts, to win

Here, because of the sentence meaning, it means to work.

The subject is plural:

• Báðir foreldrar mínir = both my parents

So the verb is also plural:

• ég vinn = I work
• þú vinnur = you work
• hann/hún vinnur = he/she works
• við vinnum = we work
• þið vinnið = you all work
• þeir/þær/þau vinna = they work

Since parents is a they subject, vinna is exactly the right form.

Because í can take two different cases depending on meaning:

• dative for location: in / inside / in
• accusative for motion into

Here the sentence says they work in the city. That is a location, not movement into it, so Icelandic uses the dative:

• í borginni = in the city

Compare:

• Þau vinna í borginni. = They work in the city.
• Þau fara í borgina. = They go into the city.

This location-vs-motion distinction is very important with Icelandic prepositions.

Because borginni contains the noun plus the definite article.

The base noun is:

• borg = city

With the definite article, it becomes:

• borgin = the city (nominative)

But after í for location, the noun must be in the dative singular:

• borg → borginni = in the city

So -inni here reflects:

1. the noun being definite (the city), and
2. the noun being in the dative singular

No, not in standard Icelandic for this meaning.

When both directly modifies a plural noun like parents, Icelandic uses the agreeing adjective:

• Báðir foreldrar mínir ...

Bæði is used in other structures, especially:

• bæði ... og ... = both ... and ...

For example:

• Bæði mamma og pabbi vinna í borginni. = Both mom and dad work in the city.

So:

• Báðir foreldrar mínir ... = correct
• Bæði foreldrar mínir ... = not the normal correct form here

Yes. That is also a good Icelandic sentence.

There is a slight difference in focus:

• Báðir foreldrar mínir vinna í borginni.
  Focuses early on both my parents as the subject.

• Foreldrar mínir vinna báðir í borginni.
  Feels a bit more like my parents both work in the city, with báðir added later for emphasis.

Both are grammatical. The original sentence is very natural and straightforward.

The basic structure is:

• Báðir foreldrar mínir = subject
• vinna = verb
• í borginni = prepositional phrase

So the pattern is essentially:

Subject + Verb + Place

That said, Icelandic word order is more flexible than English because case endings show grammatical relationships. Still, this sentence uses a very common and natural order.

Very naturally:

• Annar foreldra minna vinnur í borginni. = One of my parents works in the city.
• Foreldrar mínir vinna í borginni. = My parents work in the city.

The original sentence adds the idea that both of them do:

• Báðir foreldrar mínir vinna í borginni.

So báðir is there to make it explicit that the statement applies to the two parents together.