Sandpaper on a dowel and stropping compounds are ways to get recurve-profiles 'sorta sharp', but they typically lack bite, and at some point these methods lose their ability to bring the edge fully back on their own.
Contouring the edge of one of your stones, taking edge trailing strokes, and then stropping with compounds works well. As do edge-leading strokes on round/oval ceramic/ruby/diamond-plated honing rods, India slipstones, scythe stones, and grooved sharpening steels, providing the blade is sufficiently thinned behind the edge that these can still generate a burr/cut in a new apex. You need something hard to set that bevel, and I've not found a way to deburr/sufficiently refine such weirdly shaped blades without compounds on contoured strops quickly/reliably. Edge-leading strokes on very large, very hard, very fine slip or scythe stones (Like Translucent Arkansas) can sometimes do the trick, but it's slow, fiddly, and not guaranteed to work on all steels. Stropping on compounds tends to be a more reliable finishing technique for these blades.
You can thin them using coarser sandpaper over a compressible backing, like a wet piece of t-shirt rag, around a dowel or - better yet - oval piece of moulding. This will convex the grind up to the apex. You'll then need to cut in a bevel using a harder stone/rod, before stropping it, to get the edge desired.
Thin with sandpaper on a contourable media, bevel set on an unforgiving media, refine/zero-convex the apex on a relatively fine sandpaper (1000-2000 grit) over a contourable base, and then strop on compound. Great edge. Skip that bevel setting phase on a hard rod/stone corner, and it won't be that sharp at the end.
Periodically sharpening the spine-side of the tip, to crisp it up, also helps tip penetration for birds-beak parers. Think of them like Karambits. Keep them almost zero-thinned like scythes, but with more attention to the tip. You need them thin.
That said, I still avoid birds-beak parers like the plague. They can work for some cutting techniques, but are so difficult and frustrating to maintain geometrically and just in terms of basic sharpness, whatever they can do that more conventional blade shapes cannot virtually isn't worth it to me.
If you do get one, get a Robert Herder.
