It could be that the grass, which looks to be far too close to the peonies, is hogging all the water. Have you fertilized the peonies recently? Are they in that much sun for most of the day?

yeah, I'd say the are pretty much full sun.....these things have always been here, I'm just now starting to really pay attention to them. The owners that had the house before us must have planted them. I have never given them supplemental water or fertilizer.

How fond are you of grass? How much yard do you have over to the right of the peonies? If it were my yard, I would start about a foot to the right of the peonies and start ripping up grass, continuing to the chain link fence on the left - Voila! A lovely garden, edged with peonies!
Peonies can just exist for a long time - I found one in deep shade when we cut down a pine tree - but they won't flourish or bloom, just stay alive. They need room, they need lots of organic matter by their roots, they need fertilizer, they need water to get those enormous flowers. Your peonies are tough, growing in the grass as they are, but it's going to take a good amount of work to get them into massive-blooming mode.

Besides all that, isn't it a pain to have them marching down the middle of the yard? We had a garden in the middle of our back yard when we moved into our house, too, and it made me nuts. I finally moved everything and closed it up - much easier to mow without it.

Whatever you decide, be assured that your peonies are healthy and happy - it's just that they could be healthier and happier.

Your peonies maybe situated too deep. If the eyes are any deeper than 2 inches below soil level, they will not bloom. The plants may look really healthy, but they do not produce any buds You may need to dig them up this fall, see how deep they are. You may have to dig up the whole bed and replant at the proper depth. It is probably best to plant the peonies eyes at soil level and then put an inch of mulch over them.

I agree that the Peonies are competing with the grass. Instead of digging up the grass, check out lasagne beds! Also called "sheet composting" you can create a new bed by layering cardboard and organic materials like a pan of lasagne, and letting it all decompose while smothering and killing the grass underneath. I have created several lasagne beds and while it can take a year or two to fully decompose, the soil is now simply amazing.

Also as Kousa says the eyes may be planted too deep, and the turf grass could be contributing to that problem because it will build up roots and foliage over the peony root system.

If the peonies have been in the ground for years they will create their own level. Only newly planted ones do you need to worry about how deep they are. I agree with removing the grass and creating a flower bed.. If you do decide to dig and replant you may find that the roots may be hard and hollow or fall apart into natural clumps.