hareta hi nara, kodomotati ha kouen de genki ni asoberu.

Yes, 晴れた is the past/plain form of 晴れる. But in Japanese, the plain form of a verb can also be used to modify a noun.

So:

• 晴れる = to clear up / to be sunny
• 晴れた日 = a day that was sunny / a sunny day

In English, we would usually just say a sunny day, but Japanese often uses a verb phrase before the noun.

This does not necessarily mean the whole sentence is in the past. It is just the form used to describe 日.

なら means something like if it is..., if it’s the case that..., or when it comes to... depending on context.

Here, 晴れた日なら means:

• if it’s a sunny day
• on a sunny day

It sets up a condition for the rest of the sentence:

• If the day is sunny, the children can play energetically in the park.

With nouns and noun-like phrases, なら is very common for making this kind of conditional.

Both can mean something like if it’s a sunny day or on a sunny day, but the nuance is a little different.

• 晴れた日 = a day that turned out sunny / a day that is sunny
• 晴れの日 = a sunny day, using 晴れ as a noun

In many everyday situations, they are very close in meaning.
However, 晴れた日 feels a bit more like a natural description built from the verb 晴れる, while 晴れの日 is a more fixed noun-style expression.

Both are normal Japanese.

たち is a common suffix used to show a plural group, especially for people.

• 子供 = child / children
• 子供たち = children

Japanese often does not require plural marking, so 子供 alone could also mean children depending on context.
Adding たち makes it clearer that we are talking about multiple children.

は marks the topic of the sentence.

So 子供たちは means something like:

• as for the children
• the children

It tells us that the sentence is going to say something about the children.

In this sentence, the structure is roughly:

• As for the children, if it’s a sunny day, they can play energetically in the park.

Japanese often uses は to present the thing being talked about, even when English would just use a normal subject.

Because で marks the place where an action happens.

• 公園で遊べる = can play in/at the park

The verb 遊ぶ is an action, so the location of that action takes で.

By contrast, に is often used for:

• destination: 公園に行く = go to the park
• existence: 公園に子供がいる = there are children in the park

So here, since the children are doing the action of playing, で is the correct particle.

元気 is a na-adjective / adjectival noun meaning healthy, lively, or energetic.

To use a na-adjective adverbially, Japanese usually adds に.

So:

• 元気な子供 = an energetic child
• 元気に遊ぶ = play energetically / play actively

That is why the sentence has 元気に before the verb.

遊べる is the potential form of 遊ぶ.

• 遊ぶ = to play
• 遊べる = can play / be able to play

So the sentence is not just saying that the children play, but that they are able to play.

That fits the conditional idea:

• If it’s a sunny day, the children can play energetically in the park.

They are very similar.

• 遊べる = can play
• 遊ぶことができる = can play / be able to play

The potential form 遊べる is shorter and very natural in everyday Japanese.
遊ぶことができる is more explicit and can sometimes sound a little more formal or explanatory.

In most ordinary sentences like this one, 遊べる is the more natural choice.

The comma shows a pause between the conditional part and the main statement.

• 晴れた日なら、 = if it’s a sunny day,
• 子供たちは公園で元気に遊べる。 = the children can play energetically in the park.

Japanese commas are often used more flexibly than English commas.
This comma is helpful, but in some contexts it could be omitted.

A helpful breakdown is:

• 晴れた日なら、 = if it’s a sunny day
• 子供たちは = as for the children
• 公園で = in the park
• 元気に = energetically
• 遊べる = can play

So the sentence structure is:

[condition] + [topic] + [place] + [manner] + [verb]

This is very typical Japanese word order.
The verb comes at the end, and information like place and manner comes before it.

In isolation, 晴れる can mean to clear up or to become sunny.
But in 晴れた日, it is very natural to understand it simply as:

• a sunny day
• a clear day

So in this sentence, you do not need to force the idea of became sunny.
It is just describing the kind of day.

Yes, but the nuance would change.

• 子供たちは presents the children as the topic.
• 子供たちが would focus more directly on the children as the grammatical subject, sometimes with a more specific or contrastive feel.

In a general statement like this, は sounds very natural:

• As for children, on sunny days, they can play energetically in the park.

If you used が, it might sound more like you are specifically identifying the children as the ones who can play.

So は fits better for a broad, general statement.