Quiero cambiar la funda del cojín y también la del edredón antes de que lleguen mis padres.

Because Spanish often avoids repeating a noun that is already clear from the context.

• la funda del cojín = the cushion cover
• la del edredón = literally the one of the duvet, where la stands for la funda

So the full idea is:

• Quiero cambiar la funda del cojín y también la funda del edredón

But Spanish naturally shortens the second part to la del edredón.

Also, it uses la because funda is feminine.

del is the contraction of de + el.

• de el cojín → del cojín
• de el edredón → del edredón

This contraction is mandatory in normal Spanish, except with proper names like de El Escorial.

So here:

• la funda del cojín = the cover of the cushion
• la del edredón = the one of the duvet

Because querer is followed directly by an infinitive.

• Quiero cambiar... = I want to change...
• Quiero comer = I want to eat
• Quiero salir = I want to leave

No preposition is needed between querer and the infinitive.

Because antes de que normally triggers the subjunctive when the action is still in the future or has not happened yet.

In this sentence, the parents have not arrived yet, so Spanish uses the subjunctive:

• antes de que lleguen mis padres

This is a very common pattern:

• antes de que salgas
• antes de que empiece
• antes de que vengan

If you see antes de que, you should usually expect the subjunctive.

lleguen is the present subjunctive form of llegar for ellos/ellas/ustedes.

It is plural because mis padres is plural.

• que llegue mi padre
• que lleguen mis padres

So the verb agrees with mis padres, not with antes de que.

Yes. That is also correct and very natural.

Spanish word order is flexible here:

• antes de que lleguen mis padres
• antes de que mis padres lleguen

Both mean the same thing. The original version puts a little more focus on the arrival itself first; the second version introduces mis padres earlier.

Here funda means cover.

It is used for removable fabric covers or protective covers, for example:

• funda del cojín = cushion cover
• funda del edredón = duvet cover
• funda del móvil = phone case
• funda de almohada = pillowcase

So cambiar la funda means to change or replace the cover, usually with a clean one.

In Spain:

• cojín = cushion
• almohada = pillow

A cojín is usually decorative or used on a sofa, chair, or bed. An almohada is the thing you sleep on.

So la funda del cojín is a cushion cover, not a pillowcase.

These words are related but not the same:

• edredón = duvet / quilt / comforter
• manta = blanket
• colcha = bedspread

In Spain, if you specifically mean a duvet cover, people also often say funda nórdica.
So la funda del edredón is understandable, but funda nórdica is also very common in Spain.

That position is natural and emphasizes that this is an additional thing you want to change.

• la funda del cojín y también la del edredón

It means the cushion cover, and also the duvet cover.

Other placements are possible, for example:

• Quiero cambiar la funda del cojín y la del edredón también

But the original sounds smooth and natural.

Because Spanish normally does not use an article before a possessive adjective.

• mis padres = my parents
• mi casa = my house
• nuestro coche = our car

So los mis padres would be wrong in standard Spanish.

Yes, that is grammatically possible, but it sounds a bit different.

• Quiero cambiar la funda del cojín y también la del edredón sounds more natural when talking about the two items one by one.
• Quiero cambiar las fundas del cojín y del edredón groups them together more.

The original version is especially natural because Spanish likes to avoid repeating funda and uses la del edredón instead.

It can suggest both, depending on context.

In English, change the cushion cover often means take off the old one and put on another/clean one. Spanish cambiar la funda works the same way.

So here cambiar does not usually mean changing the cover into something different in an abstract sense; it means physically replacing it.