What is exactly やれ? Is it kind of a とか or やら？
やれ is a word that is used when you mention about two or more similar things. It has a nuance of complaining about the incident that is occurring.
= He kept calling me a fool and a dimwit.
Difference between とか and やら:
They are also used to list more than two things, but it doesn't necessarily have a complaining nuance.
= A humble man is ready to perform menial tasks such as doing chores around home or cleaning the house.
= I bought a lot of fruits - bananas, apples, and so on.
In the case of the question, やれ is used only once, but it suggests the kid's complaining attitude.
This やれ is basically an old interjection similar to "O", "Lo" or "Oh my". See this dictionary entry. As an standalone interjection, it's no longer used in modern conversations. (A similar interjection, やれやれ, is still common.)
However, it's still used in the form of
やれ A だ やれ B だ or
やれ A だの やれ B だの to list two (or more) things. This expresses A and B may be important to someone but the speaker has a negative (or at least indifferent) feeling about them. Both AだBだ and AだのBだの already have a similar function, and やれ makes it sound stronger. This is a fixed pattern you have to memorize.
Your sentence is atypical in that it has only one やれ, but I feel there is still a list (of complaints). The use of やれ implies the child is trying to keep drawing attention by complaining, and the parent has a negative feeling about it.