Anyone have any idea what model of John Deere backhoe this is? I am trying to get parts for it and there is no serial numbers or anything else on it.
You could try contacting a dealer and share your picture with them. This link will help you find a dealer near you.
Since you need parts of your John Deere backhoe, then I think you need a John Deer 1650 Parts Manual.
I found a Manual Fully Covers JD Model 1650 Backhoes, all years & serials, Complete Parts Manual (reproduction of the factory manuals) www.ebay.com/
The price is $11.45.
I have doubts that this is a Model 1650. The photo is small, but it shows that there are differences.
Compare closely the long boom shape of your John Deere Backhoe attachment with the model 1650:
The long boom shape of your Backhoe attachment is completely different, at least it looks like it in the photo you added.
For identification, we need more good quality photos of your Backhoe attachment and its dimensions.