Yes, есть is omitted when the sentence is about some quality or quantity of the the possessed thing, and not omitted when it’s about the possession itself.
У неё три́ попуга́йчика.
Having three pet parakeets is not an innate characteristic of humans, but we talk about the number here, not about the possession.

Есть is omitted with diseases for some reason:
У меня рак — I have cancer.
(«У меня есть рак» would be understood as ‘I have a crayfish’)