You probably do not want a _init with required parameters on a Node class, this makes it very difficult to instantiate, and you cannot easily wait with it. Try using _ready instead, when creating the object you will have to alter jsonString before adding it as a child
func _ready() -> void:
self.animationData = JSON.parse_string(self.jsonString)
await get_tree().create_timer(0.5).timeout # delays half a second