Learn Hardcore Russian: Родители позволяют дочери гулять одной только днём. - The parents allow their daughter to walk alone only during the day.

Учитель позволяет ученикам разговаривать. – The teacher allows the students to talk.
Родители не позволяют сыну долго сидеть в интернете. – The parents don’t allow their son to sit online for long.

Could we say своей дочери or use a pronoun like ей instead of дочери?

Yes, both are possible.

With a possessive:
- Родители позволяют своей дочери гулять одной только днём.
  This is fully correct and makes it explicit that it’s their daughter, though in context it’s usually obvious, so своей is often omitted.
With a pronoun (also dative):
- Родители позволяют ей гулять одной только днём.

All of these are grammatical:

Родители позволяют дочери гулять одной только днём.
Родители позволяют своей дочери гулять одной только днём.
Родители позволяют ей гулять одной только днём.

The difference is just how explicit you are about whose daughter you mean and whether you repeat the noun or replace it with a pronoun.

Why is the verb гулять in the infinitive form?

With позволять / позволить, the action that is allowed is normally expressed by an infinitive:

позволять кому-то что делать – to allow someone to do something

So:

Родители позволяют дочери гулять… – Parents allow (their) daughter to walk / to go out for walks…

Using a finite verb form (like гуляет) would break the construction:

✗ Родители позволяют, что дочь гуляет одной только днём. – This is unnatural / wrong.

Aspect-wise, гулять is imperfective, which fits because this is a general permission (what she is allowed to do as a rule), not a one‑time, completed action. For a one-time permission, you might see a perfective:

Родители позволили дочери один раз погулять допоздна. – They allowed her to go for a walk late once.

What does одной mean here? Does it mean “one”?

In this sentence, одной does not mean “one (single)”. It means “alone / by herself”.

So:

гулять одной ≈ “to walk alone / to go out by herself”

Compare:

Она живёт одна. – She lives alone.
Ей нравится гулять одной. – She likes to walk alone.

The underlying adjective is один, but in many contexts один / одна / одни means “alone, by oneself”, not literally “one (in number)”.

Why is it одной, not одна?

There are two different uses of this word:

Nominative predicate (“she is alone”)
- Она одна. – She is alone.
  Here одна is nominative feminine and agrees with она as a normal predicate adjective.
“Alone” with an infinitive / non-finite construction
In patterns like гулять одной, жить одной, Russians usually use одному / одной / одним, not один / одна / одни:
- Ей нравится гулять одной. – She likes to walk alone.
- Ему скучно жить одному. – It’s boring for him to live alone.
- Им пришлось идти одним. – They had to go (by) themselves.

So the “rule of thumb”:

Она одна. – she is alone (nominative).
Ей нравится гулять одной. – she likes to walk alone (this special form одной).

In your sentence, гулять одной follows that standard pattern, so одной is the form you must use.

Does только mean “only alone” or “only during the day”? How should I understand одной только днём?

In this sentence, только is understood as modifying днём, so the meaning is:

They allow her to go out alone, but only during the day (not at night).

You can think of the structure as:

гулять одной, только днём. – to walk alone, but only in the daytime.

If the speaker really wanted to say something like “they allow only her to walk during the day (others can’t)”, they’d have to phrase it differently, e.g.:

Родители позволяют гулять днём только ей. – Only she is allowed to walk during the day.

Or if they wanted to stress “only alone (not with others)”, they’d probably say:

Родители позволяют дочери гулять только одной. – They allow their daughter to walk only alone (not with friends).

Your version одной только днём is naturally interpreted as:

alone (одной)
and the allowed time is limited: only during the day (только днём).

Can I change the word order? For example: гулять только днём одной or гулять только одной днём?

Some changes are possible, but they may sound less natural or change the focus.

Родители позволяют дочери гулять одной только днём.
– Very natural. Neutral: “alone, but only during the day.”
Родители позволяют дочери гулять только днём одной.
– Grammatically possible, but less natural. The focus falls strongly on днём; одной sounds a bit “hanging” at the end.
Родители позволяют дочери гулять только одной днём.
– Can sound confusing or odd. It may be interpreted as только одной = “only to one (girl)”, i.e. “only to her”, which is not the intended meaning.
More natural alternatives that keep the same meaning:
- Родители позволяют ей гулять одной, только днём.
- Родители позволяют дочери гулять одной лишь днём.

General advice: keep одной close to гулять, and только right before what is being limited (днём in this case).

What case is днём and why is it used to mean “during the day”?

Днём is the instrumental singular of день.

Russian very often uses the instrumental (without a preposition) to mean “at / during [part of the day]”:

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

So:

Она работает днём, а отдыхает вечером. – She works during the day and rests in the evening.
Мы обычно гуляем днём. – We usually go for walks during the day.

In your sentence, днём follows this very common pattern to express “during the day”.

What is the nuance of позволять here? Is it the same as разрешать?

In this context, позволять and разрешать are very close in meaning and can often be interchanged:

Родители позволяют дочери гулять…
Родители разрешают дочери гулять…

Both mean “The parents allow/permit their daughter to go out…”.

Subtle differences:

разрешать is slightly more “formal / official”: to authorize, to give permission.
позволять can carry a nuance of letting, permitting, not forbidding, sometimes with a slight connotation of indulgence:
- Я не могу себе этого позволить. – I can’t afford / allow myself this.
- Он слишком много себе позволяет. – He allows himself too much / he oversteps.

In the family-rules context (what parents allow their child to do), both verbs are fine. The choice is mostly stylistic.

How is the whole sentence pronounced? Where are the stresses?

Here’s the sentence with stressed syllables marked in uppercase (stressed vowels in caps):

РодИтели позволяЮт дОчерИ гуляТЬ однОй тОлько днЁм.

Word by word:

РодИтели – ro-DI-te-li
позволяЮт – poz-vo-LYA-yut
дОчерИ – DO-che-ri
гуляТЬ – gu-LYATʹ
однОй – od-NOY
тОлько – TOLʹ-ko
днЁм – dNYOM

Intonation-wise, there’s usually a slight pause after гулять одной and focus on тОлько днЁм, especially if you want to stress the restriction “only during the day”.