Improve sound

Upgrade the room first. There's always something you can do to improve the room. It plays the biggest roll in the system's sound signature.

Many here know this but really hit a wall (pun intended) and never fully exploit the room's potential, myself included.

I learned by sheer accident after years of believing I had done enough. 

Room treatement is the most important upgrade for electorstatic speakers. Since I have owned ML  -> Quad -> Stax -> Sound Lab A645, I think that I have some experience with this subject. 

The room acoustics and the setup are the biggest contributors to the sound you hear. If they're not right, no gear change can help that.

Go to an audio show.  You would be surprised that really expensive speakers can  sound crappy in an untreated room.  Get some absorption panels and experiment with placement in your room.

What do you feel is lacking in your current system? Does your gut tell you something is missing? I am not a fan of AVR’s for serious audio, but the specs are better than average on the Anthem and it should drive those speakers. Still, the Parasound would probably provide a more refined sound. I agree that a decent DAC is needed regardless, Schiit Bifrost or Denafrips Ares are good starts and fit your budget. Ditch the optical connection immediately and get a BJC digital coax. Try ARC of course and room treatment if possible, but also mess around with placement, find what works best, even if it is not feasible.