At this time in the story, it seems ごんべえ was sent through the air for the third time in a row, so he is pretty much complaining about it.
The verb in the sentence is 飛ばされる, passive form of 飛ばす, to mean that ごんべえ is being sent through the air not by his own will (i.e. he is being sent by the other ducks; when working at the field; and when opening an umbrella)
ばかり has some different uses, but in this case it means “full of” (not sure how to say it in nice English) and creates this meaning of “to keep doing something”, like again and again, with a negative nuance.
いなけりゃならない is just a short form of いなければならない
In total, I would translate it to something like “Why do I have to keep being sent flying away like this?” (if someone with better English can come up with a better sentence, please do so)
But I don’t see why this would be sarcasm. He is just expressing his frustration.