Anneannem, dedemin eskiden her pazar bizi parka götürdüğünü tekrar anlattı.

Because dedemin is the subject of an embedded, noun-like clause.

In a simple sentence, you would say:

• Dedem bizi parka götürdü. = My grandfather took us to the park.

But here, that whole idea becomes the object of anlattı:

• dedemin ... götürdüğünü = that my grandfather took/used to take ...

With this very common Turkish pattern, the subject of the embedded clause usually takes the genitive ending:

• dedem → dedemin

So dedemin does not mean to my grandfather here. It marks my grandfather as the subject inside the subordinate clause.

Götürdüğünü is a nominalized verb form, meaning something like that he took or his taking.

It can be broken down like this:

• götür- = take / bring
• -dük / -dik / -duk / -dük type form = a clause-forming suffix
• -ü = third person possessive
• -nü = accusative, with buffer n

So:

• götürdüğü = that he took / his taking
• götürdüğünü = that same clause, now marked as the object of anlattı

The full chunk:

• dedemin eskiden her pazar bizi parka götürdüğünü

means:

• that my grandfather used to take us to the park every Sunday in the old days

This is a very common Turkish structure:

• [genitive subject] + [verb + -DIK + possessive + case]

Because bizi is the direct object form of we/us.

• biz = we
• bizi = us

In the embedded clause, the grandfather is taking us, so Turkish uses the object form:

• dedemin bizi parka götürdüğünü

Personal pronouns in Turkish usually show their case endings clearly, so biz would be wrong here.

Because götürmek normally takes a destination in the dative case: to the park.

• park = park
• parka = to the park
• parkta = in/at the park
• parkı = the park as a direct object

So:

• bizi parka götürmek = to take us to the park

This is just the normal case pattern used with götürmek.

They add slightly different pieces of meaning:

• eskiden = in the old days / formerly / back then
• her pazar = every Sunday

Together they show a habitual action in the past:

• eskiden her pazar = every Sunday back then

That is why the sentence is understood as used to take us every Sunday, not just a one-time event.

It means the grandmother told it again.

In this sentence, tekrar is connected to the main verb anlattı:

• Anneannem ... tekrar anlattı. = My grandmother told it again.

It does not mean the taking happened again. The repeated event is the telling, not the taking.

Because Turkish is normally a verb-final language.

The whole clause dedemin eskiden her pazar bizi parka götürdüğünü functions as the object, and then the main verb comes last:

• Anneannem, [that my grandfather used to take us to the park every Sunday], again told.

That is not natural English word order, but it is very normal Turkish structure.

A more literal shape is:

• My grandmother [my grandfather’s having taken us to the park every Sunday back then] told again.

Natural English rearranges it as:

• My grandmother told again that my grandfather used to take us to the park every Sunday.

Anneannem means my maternal grandmother.

It is made from:

• anneanne = maternal grandmother
• -m = my

So:

• anneannem = my mother’s mother / my grandma on my mother’s side

This is more specific than English grandmother, because Turkish often distinguishes:

• anneanne = maternal grandmother
• babaanne = paternal grandmother

It is not essential to the grammar, and many writers would simply leave it out:

• Anneannem dedemin eskiden her pazar bizi parka götürdüğünü tekrar anlattı.

The comma can suggest a pause or topical emphasis, something like:

• As for my grandmother, she told again...

But in standard neutral writing, the version without the comma is usually more straightforward.

So if you are learning to write Turkish, it is safest not to treat that comma as required.

Yes. Turkish word order is flexible, although some orders sound more natural than others.

For example, tekrar can move:

• Anneannem dedemin eskiden her pazar bizi parka götürdüğünü tekrar anlattı.
• Anneannem tekrar dedemin eskiden her pazar bizi parka götürdüğünü anlattı.

Both mean roughly the same thing: My grandmother told again that my grandfather used to take us to the park every Sunday.

Also, elements inside the embedded clause can move around somewhat:

• dedemin bizi eskiden her pazar parka götürdüğünü

This is still understandable, but the original order sounds quite natural.

So Turkish word order is flexible, but the most important rule remains: the main verb usually stays near the end.