I have this new growth shooting out of my peony. It is attached to a thick wood trunk underneath. Is it part of plant or a weed.

Welcome Wiggle,
In my opinion your local squirrel has buried a walnut in your nice soil around your Peony. Try to carefully dig down and remove all of the root!

Yep, looks like a walnut.

Would have been easier to pull it when it just sprouted, but it still needs to removed. Try to carefully dig/pull the invader out without damaging the peony or its roots.

I pulled all the growth out to find a huge trunk down there right in the middle. I just moved here 2 months ago and this house sat empty for 2 years. So I guess it just grew unchecked. I am so scared I will hurt my peony try to dig it out, but will give it a go

Look like a walnut tree, get it out.