Why do we put an article in
We took photos of the red squirrels in St James Park.
and not
There are dogs in the garden.
In the first sentence I understand the use of "the" it is not all red squirrels but the ones we can see in St James Park, the definite article is justified.
But in the second sentence it is the same for me: it is not all dogs but these that we can see in the garden and it is a specific garden the one I am talking about: so logically we should add "the" before "dogs".
the refers to something that is either explicitly or implicitly defined outside the context of the current sentence, or within a sub-clause of it.
When we use there is/are in the sense you are using it, we are generally explaining that something indefinite exists or is available:
There is ice in the freezer
If you want to talk about some particular ice, whether explicitly or implicitly defined, we say
The ice is in the freezer
If you are introducing a reference to some dogs that you can see, you don't use the, because the dogs are defined within the sentence, rather than outside. You say
I can see some dogs.
If the dogs that you can see are in the garden, you can say:
I can see some dogs. The dogs are in the garden.
This is a legitimate use of the, because the dogs you are referring to are explicitly defined in the previous sentence (outside the current sentence).
The dogs [that I can see] are in the garden.
This is also a legitimate use of the, because the dogs you are referring to are explicitly defined in a sub-clause of the sentence [that I can see].
The dogs could be defined implicitly too. For example, if you are talking to somebody who knows that you have several dogs, you could say:
The dogs destroyed some cushions yesterday.
The listener will understand the implicit reference to your dogs.
There are [the] dogs [that are in the garden] in the garden.
Looking at your first sentence,
We took photos of the red squirrels [that are in St James's Park]
the red squirrels are explicitly defined in a sub-clause of the sentence - "that are in St James's park".
It is expected that there are always red squirrels in St James's Park, so the red squirrels are the ones in St James's Park. (I can infer this even though I'm not familiar with London simply because of the grammar of the sentence.)
However, it is not expected that there are always dogs in the garden. "There are (some) dogs in the garden."
If they are your dogs, then you could say "There are the dogs in the garden" out of context, with "the dogs" being understood to mean yours. This sentence would also indicate a bit of surprise due to "there are". "I can't find Spot and Rover. Oh, there are the dogs in the garden!" Otherwise, you'd just say "The dogs are in the garden."
If they are not your dogs, then "There are dogs in the garden." means "There are some strange/unfamiliar dogs in the garden."
