После дождя в парке прохладно.

Because после requires the genitive case.

• Dictionary form: дождь = rain
• Genitive singular: дождя

So после дождя means after rain / after the rain.

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после фильма = after the movie

Because в is being used to mean in, expressing location, and after в with location Russian uses the prepositional case.

• Dictionary form: парк
• Prepositional singular: в парке

Compare:

• в парке = in the park
• в парк = into the park

So here, since the sentence says where it is cool, not movement into the park, в парке is correct.

In present-tense Russian, the verb to be is usually omitted.

English needs it is cool.
Russian can simply say прохладно.

So the sentence is completely normal and complete without a verb like is:

• В комнате тепло. = It is warm in the room.
• На улице холодно. = It is cold outside.
• В парке прохладно. = It is cool in the park.

Here прохладно is not an adjective describing a noun. It is a predicative word or state word, used to describe an overall condition or environment.

It means something like:

• it is cool
• it feels cool

Compare:

• прохладно = it is cool
• прохладный день = a cool day
• прохладная вода = cool water

So in this sentence, Russian is not saying that some noun is cool. It is saying that the atmosphere in the park is cool.

Because there is no noun here for an adjective to agree with.

Похладный would need a noun:

• прохладный парк = a cool park
• прохладный вечер = a cool evening

But После дождя в парке прохладно is an impersonal statement about the general conditions, so прохладно is the natural form.

It can mean either one, depending on context.

Russian has no articles like a or the, so дождя does not tell you by itself whether the speaker means:

• rain in general
• a specific rain that just happened

Usually the situation makes it clear. In this sentence, many English speakers would naturally understand it as after the rain or after it rains / after rain depending on context.

Yes. Russian word order is flexible.

This version is neutral and natural:

• После дождя в парке прохладно.

But you could also say:

• В парке после дождя прохладно.
• Прохладно в парке после дождя.

The basic meaning stays the same, but the emphasis changes a little.

Very roughly:

• После дождя first puts the time condition first
• в парке first emphasizes the place
• прохладно first emphasizes the feeling or condition

Yes. It is an impersonal construction.

There is no grammatical subject like the park doing something. Russian often uses this kind of sentence for weather, temperature, physical conditions, and general states:

• Холодно. = It is cold.
• Темно. = It is dark.
• Тихо. = It is quiet.
• В парке прохладно. = It is cool in the park.

English often uses a dummy subject it, but Russian does not need one.

The main stresses are:

По́сле дождя́ в па́рке прохла́дно.

A few helpful notes:

• после → stress on the first syllable: По́сле
• дождя → stress on the last syllable: дождя́
• парке → stress on the first syllable: па́рке
• прохладно → stress on the second syllable: прохла́дно

A rough English-style approximation would be:

POS-le dozh-DYA v PAR-ke pro-KHLAD-na

Not exact, of course, but useful as a starting point.