Après la dispute, Paul et Marie parlent calmement.

Because Paul et Marie is a plural subject: two people are doing the action.

The verb parler in the present tense changes like this:

• je parle
• tu parles
• il / elle parle
• nous parlons
• vous parlez
• ils / elles parlent

Since Paul et Marie = they, French uses parlent.

Because French does not need a subject pronoun when the subject is already clearly stated.

So:

• Paul et Marie parlent calmement. = Paul and Marie speak / are speaking calmly.

You would only use ils if you were replacing Paul et Marie:

• Paul et Marie parlent calmement.
• Ils parlent calmement.

French does not normally say both together the way English sometimes can for emphasis.

Après is a preposition meaning after.

In Après la dispute, it introduces a time expression:

• après = after
• la dispute = the argument / quarrel

So the phrase means after the argument.

French often begins a sentence with a time phrase like this:

• Après le dîner, nous sortons.
• Après la dispute, Paul et Marie parlent calmement.

The comma separates the opening time phrase from the main clause.

• Après la dispute, = introductory time phrase
• Paul et Marie parlent calmement. = main clause

This is very natural in French, just as in English:

• After the argument, Paul and Marie speak calmly.

In short sentences, the comma is sometimes a style choice, but here it is completely normal and helpful.

Because French adverbs often come after the conjugated verb, especially adverbs of manner such as calmly, slowly, seriously, etc.

So:

• parlent calmement = speak calmly / are speaking calmly

Compare:

• Il répond poliment. = He answers politely.
• Elle travaille rapidement. = She works quickly.

English often allows more flexibility, but in French, putting calmement after the verb is the standard position here.

Calmement is an adverb formed from the adjective calme.

Very often, French makes adverbs by adding -ment to the feminine form of an adjective, or to the adjective itself if the masculine and feminine are the same.

Here:

• calme = calm
• calmement = calmly

Other examples:

• rapide → rapidement
• heureux → heureusement
• doux → doucement

So calmement describes how they are speaking.

It can mean either one, depending on context.

French present tense often covers both:

• they speak
• they are speaking

So Paul et Marie parlent calmement could mean:

• they speak calmly
• they are speaking calmly

Because the sentence begins with Après la dispute, many learners will naturally understand it as a scene happening now or at that moment: After the argument, Paul and Marie are speaking calmly.

Because dispute is a feminine noun in French.

French nouns have grammatical gender, so you must learn the noun together with its article:

• la dispute

That is why you say:

• après la dispute

and not après le dispute.

For learners, it is best to memorize nouns with their article:

• la dispute
• le problème
• la conversation
• le débat

Not always.

In this sentence, la dispute usually means an argument, a quarrel, or a fight in words between people.

In English, dispute can sound more formal or abstract, like a disagreement between groups or over facts. French dispute very often refers to a personal argument.

So here, a natural English sense is:

• After the argument...

rather than a very formal After the dispute...

Yes, that is grammatically possible, but it sounds a little different in focus.

• Après la dispute, Paul et Marie parlent calmement.
  This puts the time frame first: After the argument...

• Paul et Marie parlent calmement après la dispute.
  This keeps the main subject first and adds the time information later.

Both are correct. The version with Après la dispute at the beginning gives slightly more emphasis to the sequence of events.

This is a very common French question.

In the ils / elles form of many regular -er verbs, the ending is written -ent, but it is usually silent.

So:

• parle = pronounced like parl
• parles = same pronunciation
• parlent = same pronunciation

That means:

• je parle
• tu parles
• il parle
• ils parlent

all sound the same in normal speech.

French spelling often shows grammar even when pronunciation does not change.

Yes, if it is already clear that ils refers to Paul et Marie.

For example:

• Après la dispute, Paul et Marie se retrouvent dans la cuisine. Ils parlent calmement.

But if you are introducing them for the first time, using their names is clearer:

• Après la dispute, Paul et Marie parlent calmement.

So both are correct; it depends on whether the people have already been identified.