Mi vidas kurantan hundon en la parko.

Because hundon is the direct object of vidas (I see ...), and in Esperanto the direct object usually takes -n.

Since kurantan describes hundon, it must agree with it. Adjectives and adjective-like words in Esperanto match the noun in:

• number: singular/plural
• case: whether it has -n

So:

• hundo = a dog
• hundon = a dog (as direct object)
• kuranta hundo = a running dog
• kurantan hundon = a running dog (as direct object)

Kurantan comes from the verb kuri = to run.

The part -ant- is the present active participle, meaning doing the action right now / in an ongoing way.

So:

• kuri = to run
• kuranta = running
• kurantan = running (with -n added to agree with hundon)

In this sentence, kurantan hundon means a dog that is running.

Because kuranta must agree with hundon.

Since hundon has -n, the describing word must also have -n:

• correct: kurantan hundon
• incorrect: kuranta hundon

This is a very common Esperanto rule: adjectives and participles match the noun they describe.

Here it works like an adjective.

Even though it comes from the verb kuri, the participle kuranta behaves like a describing word:

• kuranta hundo = a running dog

So in this sentence, kurantan is not the main verb.
The main verb is vidas.

Structure:

• Mi = subject
• vidas = main verb
• kurantan hundon = object phrase

Because en la parko shows location: the dog is in the park.

In Esperanto, after a preposition, -n can sometimes show direction toward a place rather than location.

So:

• en la parko = in the park
• en la parkon = into the park

Here the sentence describes where the dog is being seen, not movement into the park, so en la parko is the normal form.

It most naturally tells you the setting of the situation: the dog is being seen in the park.

In practice, it can often be understood as connected with the whole scene:

• I see a running dog in the park.

Context usually makes it clear. If necessary, Esperanto can be rephrased to remove ambiguity.

For example:

• En la parko, mi vidas kurantan hundon.
• Mi vidas hundon, kiu kuras en la parko.

But your original sentence is normal and natural.

Because Esperanto has no indefinite article.

English distinguishes:

• a dog
• the dog

Esperanto only has a definite article:

• la = the

So:

• hundo / hundon can mean a dog
• la hundo / la hundon means the dog

That is why hundon by itself is enough.

Because Esperanto uses la for a specific, identifiable thing.

So en la parko means in the park.
If you said en parko, that would sound more like in a park or in parkland, depending on context, and is less likely in a sentence like this.

In many ordinary situations, if both speaker and listener can identify the place, la is natural.

Yes. That is also grammatical.

Esperanto word order is fairly flexible, and both of these are possible:

• Mi vidas kurantan hundon en la parko.
• Mi vidas hundon kurantan en la parko.

The first one puts the description running before the noun, which is very common.
The second puts it after the noun, which is also fine.

Both can mean essentially the same thing.

Yes, but that changes the meaning slightly.

• Mi vidas kurantan hundon en la parko. = I see a running dog in the park.
• Mi vidas la kurantan hundon en la parko. = I see the running dog in the park.

Adding la makes the dog specific or already known.

No, Esperanto word order is more flexible than English because endings show grammatical roles.

In this sentence:

• Mi is the subject
• vidas is the verb
• hundon is the object because of -n

So Esperanto can move parts around more easily than English, though some orders sound more natural than others.

For example, these are possible:

• Mi vidas kurantan hundon en la parko.
• En la parko mi vidas kurantan hundon.

The original order is neutral and natural.

You would make both the noun and its modifier plural.

• kurantaj hundoj = running dogs
• Mi vidas kurantajn hundojn. = I see running dogs.

Notice the agreement:

• singular: kurantan hundon
• plural: kurantajn hundojn

Both words show plural, and both take -n if they are the direct object.