Les enfants regardent le feu d’artifice avec leurs parents.

Les means the, while des means some.

So les enfants refers to specific children already known from the context: the children.
If you said des enfants regardent le feu d’artifice, it would mean some children are watching the fireworks.

Because the subject is les enfants, which is third person plural: they.

The verb regarder means to watch / to look at, and in the present tense:

• je regarde
• tu regardes
• il/elle regarde
• nous regardons
• vous regardez
• ils/elles regardent

So with les enfants, you use regardent.

In French, the simple present often covers both ideas:

• they watch
• they are watching

So Les enfants regardent le feu d’artifice can naturally mean The children are watching the fireworks.

If you really want to emphasize the ongoing action, French can also use être en train de:

• Les enfants sont en train de regarder le feu d’artifice.

But in everyday French, the simple present is usually enough.

French uses feu d’artifice as a fixed expression.

Literally, it is something like firework display or artificial fire, but the natural meaning is fireworks.

A key difference from English is that French often uses this expression in the singular:

• un feu d’artifice = a fireworks display
• le feu d’artifice = the fireworks / the fireworks display

So even though English often uses the plural-looking word fireworks, French commonly uses the singular phrase feu d’artifice.

Because it comes from de + artifice.

In French, de becomes d’ before a vowel sound:

• de + artifice → d’artifice
• de + eau → d’eau
• de + école → d’école

This is called elision.

Because the owner is plural.

• son / sa / ses = his / her / its
• leur / leurs = their

Here, the subject is les enfants (the children), so the parents belong to them, not to just one person. That is why French uses leurs.

Because parents is plural.

In French, possessive adjectives agree with the thing possessed, not with English-style distinctions like his vs her.

Compare:

• leur parent = their parent
• leurs parents = their parents

Since parents is plural, you need leurs.

It usually means their parents in a natural, general sense, often understood as each child with his or her own parents, depending on context.

French does not always make that distinction explicitly.
So Les enfants regardent le feu d’artifice avec leurs parents is a normal way to say that the children are watching with their parents, without over-explaining whose parents belong to whom.

Because parents normally means mother and father / parents.

French uses:

• un parent = a parent / relative depending on context
• les parents = the parents

So leurs parents means their parents.

That is the most natural word order in French.

The basic structure is:

• Les enfants = subject
• regardent = verb
• le feu d’artifice = direct object
• avec leurs parents = extra information (with their parents)

French word order is often similar to English here:

• The children
  • are watching
    • the fireworks
      • with their parents

Yes. Regarder is the normal verb for to watch or to look at something.

Examples:

• regarder la télévision = to watch TV
• regarder un film = to watch a movie
• regarder le feu d’artifice = to watch the fireworks

A related verb is voir (to see), but that is different:

• voir = to see
• regarder = to look at / watch

So here regardent is exactly the right choice.

A careful approximate pronunciation is:

Lay-zahn-fahn ruh-gar-duhn luh fuh dar-tee-fees ah-vek luhr pah-rahn

A few useful pronunciation points:

• les enfants has a liaison: the s in les sounds like z before enfants
• final -ent in regardent is silent
• final s in parents is silent
• the en/an sound in enfants and parents is nasal

A more IPA-style version is:

[lez ɑ̃fɑ̃ ʁəɡaʁd lə fø daʁtifis avɛk lœʁ paʁɑ̃]

Because it is a noun phrase functioning as the direct object: the fireworks / the fireworks display.

French generally requires an article where English sometimes does too, and often even where English might omit one. Here, since it means the fireworks, le is needed:

• les enfants regardent le feu d’artifice

Without the article, the sentence would sound incomplete or ungrammatical in normal French.