Learn Hardcore Norwegian: Under foreldrepermisjonen går hun ofte en rolig tur med babyen. - During parental leave she often goes for a calm walk with the baby.

Norwegian often starts with a time/place phrase to set the scene. When you put something other than the subject first, Norwegian uses V2 word order (the finite verb must be in position 2).
So Under foreldrepermisjonen takes position 1, then the verb går must come next, and then the subject hun:

Under foreldrepermisjonen (1) går (2) hun …

If you start with the subject, you get:

Hun går ofte en rolig tur med babyen under foreldrepermisjonen.

What does under mean here—does it literally mean “under”?

Here under means during / in the period of. It’s common with time periods:

under ferien = during the holiday
under møtet = during the meeting
It can also mean physical “under,” but context makes it clear this is time.

Why is it foreldrepermisjonen (definite form) and not foreldrepermisjon?

The definite form -en often refers to a specific, known instance: her parental leave (the one she is on). In context, it’s typically a particular leave period, so definite sounds natural:

under foreldrepermisjonen = during the parental leave (hers/the relevant one)

You can also see indefinite in more general statements:

Under foreldrepermisjon kan man… = During parental leave (in general), one can…

Is foreldrepermisjon one word? How are long words like that built?

Yes—Norwegian commonly forms compound nouns as one word.
foreldrepermisjon = foreldre (parents) + permisjon (leave).
Then add definiteness: foreldrepermisjon-en = foreldrepermisjonen.

Why is the verb går placed before hun?

Because of V2: the finite verb must be the second element in a main clause. Since Under foreldrepermisjonen is first, the verb går must come next, forcing hun after it:

Under foreldrepermisjonen går hun…

This is similar to English in questions (“Yesterday did she…?”), but in Norwegian it happens in regular statements too.

What role does ofte play, and where can it go in the sentence?

ofte is an adverb meaning “often.” In Norwegian main clauses it commonly appears after the subject (or after the verb+subject if there’s inversion), and usually before the object/complement:

Under foreldrepermisjonen går hun ofte en rolig tur…

You can move it for emphasis, but the most neutral positions are:

Hun går ofte en rolig tur…
Ofte går hun en rolig tur… (more emphasis on “often,” and triggers inversion)

Why is it en rolig tur and not en rolig gåtur?

tur is a broad word meaning a trip/outing/walk. With gå and context, en tur often naturally means “a walk.”
gåtur is more explicitly “a walking trip / walk” and is also common:

Hun går ofte en rolig gåtur med babyen. (very clear it’s a walk)

Both work; tur is just more general and very idiomatic.

Is går en tur the same as English “walks a walk”? Why is tur used like an object?

It’s an idiom: gå en tur = “go for a walk / take a walk.” Norwegian treats en tur as a natural complement of gå in this expression.
Another very common option is:

tar en tur = “takes a walk/trip” (also idiomatic)

So:

Hun går en tur and Hun tar en tur are both normal; they just use different verbs with the same noun.

What’s the difference between går … en rolig tur and tar … en rolig tur?

Both are common. Often:

gå en tur focuses a bit more on the act of walking/going.
ta en tur is a very general “go for a (short) trip/outing,” and can sound slightly more like “let’s head out for a bit.”

In many everyday contexts, they’re interchangeable:

Jeg tar en tur på butikken. (I’ll pop to the shop.)
Jeg går en tur med hunden. (I’m going for a walk with the dog.)

Why is it med babyen (definite) and not med en baby?

babyen = “the baby” (a specific baby—presumably hers / the one already known). That’s the natural choice here.
med en baby would mean “with a baby” (some unspecified baby), which usually doesn’t fit this context.

You could also say:

med barnet = with the child (slightly more general, and can imply an older baby too)

Could I say på foreldrepermisjon instead of under foreldrepermisjonen?

Yes, and it’s very common. The nuance changes slightly:

på foreldrepermisjon = on parental leave (your status)
under foreldrepermisjonen = during the parental leave period (time frame)

Both can work in many sentences:

På foreldrepermisjon går hun ofte en rolig tur med babyen.

What’s the most natural pronunciation focus in this sentence?

A few useful points:

foreldrepermisjonen is long; many speakers give clear stress early: FOREL-dre-per-mi-sjo-nen (stress patterns can vary by dialect, but the first part is prominent).
rolig is typically RO-lig (the g is not a hard “g” like in “go”; it’s softer, and in some dialects can be very soft or almost disappear).
tur has a rounded u sound (not like English “tour”).
babyen ends with the definite -en: BA-by-en (often flowing together).