Nawet deszcz nie przeszkadza nam spacerować po parku.

Nawet means even.

At the start of the sentence, it emphasizes the surprising part:

• Nawet deszcz... = Even the rain...

It tells you that rain is something you might expect to be a problem, but in this case it is not.

Polish word order is flexible, so nawet can sometimes appear in other positions, but here sentence-initial position sounds natural and gives strong emphasis.

Deszcz is in the nominative case because it is the subject of the sentence.

• deszcz = rain
• deszcz nie przeszkadza = the rain does not bother / prevent

Even though the sentence contains nie, that does not change the subject into another case. The subject stays in the nominative.

Because the verb przeszkadzać normally takes the dative case for the person affected.

• my = we
• nam = to us / us (dative form)

So:

• deszcz nie przeszkadza nam = the rain does not bother us / does not prevent us

This is a very common pattern:

• przeszkadzać komuś = to bother / interfere with someone
• komuś means the person is in the dative

A very useful pattern is:

• przeszkadzać komuś w czymś = to hinder someone in something
• przeszkadzać komuś coś robić = to hinder someone from doing something

In this sentence you have the second pattern:

• przeszkadza nam spacerować = prevents us from walking / stops us from walking

So the structure is:

• deszcz — subject
• nie przeszkadza — does not hinder
• nam — us
• spacerować — to walk / stroll
• po parku — around the park

Because after przeszkadzać you can use an infinitive to describe the action that is being hindered.

• przeszkadza nam spacerować = it prevents us from walking
• literally: it hinders us to walk

This is normal in Polish. English usually prefers from walking, but Polish often uses the infinitive directly.

You could also express the idea with a noun form:

• nie przeszkadza nam w spacerowaniu po parku

That is also correct, but the version with the infinitive is very natural.

Po parku means around the park or through the park.

The preposition po with the locative case often suggests movement over or around an area:

• spacerować po parku = to walk around the park

By contrast:

• w parku = in the park

So:

• spacerować w parku = to walk in the park
• spacerować po parku = to stroll around the park

Both can be possible depending on what you want to emphasize, but po parku often gives a stronger sense of moving around within the park.

Because po requires the locative case here.

The noun park changes like this:

• nominative: park
• locative: parku

So:

• po parku = around the park

This is a common preposition-case combination to remember:

• po + locative

Because przeszkadzać is imperfective, while przeszkodzić is perfective.

• przeszkadzać = to be bothering / to hinder
• przeszkodzić = to hinder successfully, to get in the way once, to stop

In this sentence, the speaker is describing a general present situation:

• Even the rain doesn’t stop us from walking in the park

For general or ongoing present meaning, Polish normally uses the imperfective verb:

• przeszkadza

The perfective przeszkodzi would usually refer to the future:

• Deszcz nam nie przeszkodzi spacerować po parku = The rain will not stop us from walking in the park

Yes, Polish word order is fairly flexible, but changing it changes the focus or emphasis.

The original:

• Nawet deszcz nie przeszkadza nam spacerować po parku.

This strongly emphasizes even the rain.

Other possible orders include:

• Deszcz nawet nie przeszkadza nam spacerować po parku.
• Nam nawet deszcz nie przeszkadza spacerować po parku.

These are grammatical, but they sound different in emphasis. For learners, the original sentence is a very natural and clear version.

It can match all of those depending on context.

• spacerować often means to walk for pleasure
• it is often closer to stroll or take a walk
• in simpler translations, it is often just walk

So spacerować po parku is something like:

• walk around the park
• stroll through the park
• take a walk in the park

All are reasonable depending on style and context.