Després de descansar a casa, vull tornar a caminar al parc.

Després de means after.

In Catalan, the idea is built as:

després de + noun / infinitive

So:

• després de la feina = after work
• després de descansar = after resting

The de is just part of the fixed expression. English uses after, but Catalan usually uses després de.

Because after després de, Catalan normally uses an infinitive when the action is general or when the subject is understood.

So:

• després de descansar = after resting
• després de menjar = after eating
• després de parlar = after speaking

This works a lot like English after + -ing, but Catalan uses the infinitive instead of a gerund here.

A casa is the usual way to say at home or home in Catalan.

It is an idiomatic expression:

• ser a casa = to be at home
• anar a casa = to go home

If you say a la casa, it usually sounds more like to the house or refers to a specific house as a building, not the general idea of home.

So in this sentence, descansar a casa naturally means to rest at home.

Vull means I want.

It is the 1st person singular form of the verb voler (to want):

• jo vull = I want
• tu vols = you want
• ell/ella vol = he/she wants

In the sentence:

vull tornar a caminar al parc

it means I want to go back to walking / I want to walk again in the park.

Because Catalan often drops subject pronouns when the verb form already shows who the subject is.

So:

• vull already clearly means I want
• jo vull is possible, but usually used only for emphasis or contrast

For example:

• Vull descansar. = I want to rest.
• Jo vull descansar, però ell no. = I want to rest, but he doesn’t.

In your sentence, leaving out jo is completely normal and natural.

Tornar a + infinitive means to do something again or to go back to doing something.

Examples:

• tornar a veure = to see again
• tornar a començar = to start again
• tornar a caminar = to walk again

So:

vull tornar a caminar

means I want to walk again or I want to start walking again.

This is a very common Catalan structure.

Because the verb tornar uses the pattern:

tornar a + infinitive

That a is required in this structure.

Compare:

• Torno a llegir aquest llibre. = I’m reading this book again.
• Ha tornat a ploure. = It has rained again.
• Vull tornar a caminar. = I want to walk again.

So the a is not optional here.

Caminar means to walk.

It focuses on the physical act of walking.

Passejar often means to stroll, to take a walk, or to go for a walk, and can sound a bit more leisurely.

So:

• caminar = walk
• passejar = stroll / go for a walk

In many contexts they overlap, but they are not always exactly the same. In your sentence, caminar al parc suggests the speaker wants to walk in the park again.

Because a + el contracts to al in Catalan.

So:

• a + el parc → al parc
• a + el cinema → al cinema

This is just like:

• de + el → del

So al parc means to the park or sometimes in/at the park, depending on context. In this sentence, it is naturally understood as walking in the park or at the park.

It is natural and usually recommended, because the sentence begins with an introductory phrase:

Després de descansar a casa, ...

That opening part sets the time or circumstance for the main clause. The comma helps separate it from the main idea:

• Després de descansar a casa, vull tornar a caminar al parc.

You may occasionally see similar sentences without a comma in informal writing, but the comma is the clearest and most standard choice here.

Yes. Catalan allows some flexibility in word order.

For example, you could also say:

Vull tornar a caminar al parc després de descansar a casa.

This has the same basic meaning. The difference is mainly in emphasis:

• Després de descansar a casa, ... emphasizes the time/circumstance first
• Vull tornar a caminar al parc ... emphasizes the main action first

Both are grammatical. The original version sounds very natural.