There's a sofa, two armchairs, a TV and a big cage for our parrots.

This is correct.

There's is a contraction and can mean there is, there has or there are. In this case, it stands for there are. As Snailplane mentions, the there are case has become standard in modern informal English, despite the fact that apostrophe-s isn't a sensible contraction for are. It's inappropriate in formal English to use there's to mean there are, but the same applies to all contractions as a general rule, because contractions are informal.

Why has this happened? Because there's is so ubiquitous and easy to say that this is now how a broad swath of native English speakers naturally talk. And we write the way we talk, so the new meaning of there's is valid in written English as well. Most dictionaries and other English references have not caught up with the new usage, so you won't find an entry in them explaining the there are case. Such materials are (in the case of English) by definition reactive; they don't dictate rules and meanings, they document them, and that takes time and effort.

What happens when we un-contract there's? Let's set aside the there has case, since it's not relevant to the question.

Normally, we conjugate the verb based on the subject. But, there is a dummy subject and can be either singular or plural, so for there is X, we must examine X to determine the plurality of is, as if it were the subject of the sentence. If X is a lone noun, the decision is easy. Use is for singular nouns and are for plural ones.

With lists, we decide how to conjugate based on the list's construction. For a conjunctive list (formed with and), then the correct conjugation is are. For a disjunctive list (formed with or), then the verb is pluralized based on the adjacent list item. For example, there are two small ones or one big one and there is one big one or two small ones. Aside from the exception in the case of or groups, whether the items are plural doesn't matter. A list of singular nouns grouped with and always calls for a plural verb.

These are the standard rules for plural conjugation, and they're unaffected by the new wrinkle in the meaning of there's. There are X [singular] or Y, there are [singular noun], there is A and B, and there is [plural noun] are all ungrammatical.

So, for our example, removing the apostrophe-s yields this:

There are a sofa, two armchairs, a TV and a big cage for our parrots.

Let's examine the proffered amended version.

There's a sofa, there are two armchairs, there's a TV and a big cage for our parrots.

Breaking up the statement into a list of the form there's X, there's Y, and there's Z is valid, though more verbose. But what about each independent clause?

• There's a sofa - Correct.
• there are two armchairs - Correct.
• there's a TV and a big cage for our parrots - Correct. But, this is the same usage of there's as in the original; it stands for there are, because the two items are grouped together with and.

This version is correct, if a little bit awkward sounding, but it still uses there's in the same way as the original. Which is, of course, also correct.