Hi Elizabeth and thanks for your question about your Peonies. Without a picture, I can't be sure, but what you are describing as "tar looking spots" sounds to me like... insect poop! Particularly flies leave a tiny spot of tar-like substance and if you have any kind of population of flies, they might particularly like your peonies!

If this doesn't sound right, send in a picture and I'll try to help more.

Hi, thank you for getting back to me. I’ve got a feeling it’s some kind  of rust. 

Hi Elizabeth- Thanks for your picture of the leaves of your peony.
This looks like Peony Leaf Blotch which is a fungus which occurs when the temperatures and rainfall are just about what we have been having. It will be important to remove the old tops in fall or early spring before new growth appears and cut old stems at ground level and destroy them. That should keep this under control but you can also pick off the worst of the leaves now to keep air flowing around the plants. The hot weather we are about to have should help as well.
https://pnwhandbooks.org/plantdisease/host-disease/peony-paeonia-spp-leaf-blotch