You do not give any sizes, but if the bore is large enough to get in with a small threading tool try doing an internal groove at the parting position and then part down into this.
We use Iscar HeliGrip inserts for parting and by cutting the groove first we can get a small chamfer on the parted hole with a very tiny ridge around it. A quick swipe across some emery cloth and the ridge is gone.
If the hole is too tiny for doing the internal groove we use two parting tools; a carbide one to go nearly all the way down, nearly being within 5 thou or so of the piece actually parting off, then we finish with a high speed tool that has an extreme angle on it so it just slices the piece of with a burr so small it can be taken off with a quick swipe using a deburring tool.
An open mind is a virtue...so long as all the common sense has not leaked out.