I think your confusion stems from the usage of plural "parlamenti".
But the key to resolve the confusion is in the seemingly minor i, in "i Parlamenti".
In the following I explain why.
You are interpreting the phrase as:
In democrazia ci sono parlamenti che decidono ed altri parlamenti che non decidono.
But in this case, if we were to leave out the second part of the phrase (as in your interpretation), we would be left with:
In democrazia ci sono parlamenti che decidono.
This is a perfectly valid phrase, and its meaning is the one you have thought about.
But that's not your phrase.
Slightly edited for clarity, your phrase is:
In democrazia ci sono i parlamenti che decidono.
Here the plural and the definite article hint to the fact that the statement refers to all (parliamentary) democracies, not to any one in particular.
I, personally, agree that this is not the most elegant phrase and it certainly suggests an informal register.
Another way to express the same meaning could be:
In democrazia sono i parlamenti a decidere.