I've thought about this often because there remains the question of defining what exactly a reader is. Reading has classically been associated with the senses of touch and sight, even before braille there was a tactility to how we wrote due to the prevalent materials for tablets in the west and near east being either a hardened wax, clay or stone. So by that narrow definition an audiobook is not reading, it engages us aurally.
However that's not the end of what reading means, and since a reader is that which performs the action of reading its not the terminus of what a reader is as well. There's also the sense of reading via gathering visual information; one often talks of reading you like a book, or reading the weather by glancing up at the sky and foreseeing a storm. In that sense an audiobook is still not reading, we haven't yet escaped the tyranny of sight. Where we do find our liberation is when we use reading to mean interpretation and therein I find the most useful way of defining reader in regards to what media of information count as reading.
If we take into account the active mental pursuit of interpretation as being the core of reading, then audiobooks cross over into the category of reading. If we take reading to just being cognoscent that there are letters on a page then reading becomes almost a binary act. I cannot understand a single pictogram of Cantonese but I could "read" the entire works of Mencius in the sense that I am aware of the symbols and could reproduce them given materials. But if we take it to the realm of actively engaging with the matter and utilizing it then an audiobook can be the equal of any written word. If anyone argues with me on that then they need to first address the proud history of the lecture in education. All that given however we understand that the original question is in itself deficient. Do audiobooks count as reading? It focuses on the object, not the subject. Its a reification of reading, and that I care not for. What are you doing with the audiobook? Are you able to make notes or cogitate it as you encounter each new word? Or are you using it as a background noise while you fluff your pillows? The written word is harder to multitask on, but I've managed to read a book remaining distracted the whole time. If your chief concern with media is the object itself rather than your dialoguing with it then I'd argue that any claim to reading could be superficial at best.