In 有一只小鸟真可怜，它在树枝上冷得发抖 why is 有 at the start? So I am studying this children´s story, and I can´t find a proper grammar explanation.

有一只小鸟真可怜，它在树枝上冷得发抖。

So here I assume 有 indicates the bird´s existence, however in every explanation about 有 grammar I could find they would structure the sentence so that 有 would come after the place and before the subject. (Time/place) + 有 + (subject).
I couldn´t find any grammar that explained this particular kind of compound sentence, nor any other possible explanation. Any ideas?
So here I assume 有 indicates the bird‘s existence

Correct. 有 is frequently used as the equivalent of &quot;there is&quot; in English. The typical construction is

place + 有 + noun
Example: 我家里 (place) 有 (existence) 很多椅子 (subject) -&gt; There's many chairs in my house

Then you can also omit the place to express just generic existence, as in your sentence:

有一只小鸟真可怜 (nowhere in particular) there's a little bird which is really pitiful

This construction can be productively used in story-telling, where the exact location of the subject is not very relevant. Same as in English.
If you add a time complement instead of a location, you get the classical story introduction:

很久以前有一只小鸡~ once upon a time (nowhere in particular) there was a small chicken...