You're in for a world of fun.
The visible threads prove that the cup was threadded in originally. But over time something has bonded them together. That could be galvanic corrosion, or simply dirt, or a crossed/damaged thread.
Start by double-checking the thread. Put the lockring back on to confirm which way is on/off. Add penetrating oil and heat to help it soak in.
You are using the correct tool to interface with the flanges, but if its old then the edges might be worn and slightly curved. Clean them with a wire brush, and maybe score in the corner with a triangular file. Then retry the park tool. If that doesn't work you can try a large spanner but I find they're too thick and twist.
Some have suggested putting the whole frame horizontally into a secured bench vise, which might help grip the little lip.
For a non-cottered crank I made a spacer that lets you refit the crank bolt and retain the park tool on the flanges. The park tool can handle some light tapping with a hammer, but not a lot. If it starts denting then you're hitting it too hard.
Use heat on the frame, ice on the cup, and a lot of cursing can help.
If you can't unscrew either side, then it might be a lost cause.
If you're planning on installing a cartridge BB then the axle and cups can be destructively removed. You need to get the axle out, which requires a cup to come out. The cups are hardened steel so they don't drill or cut easily. You might consider a carbide drill bit to put two holes into the cup, and then try a pin spanner to turn it.
You could also try welding one cup to the axle, and then using the axle to turn the cup. Note the heat of welding will expand the cup even harder into the frame, so pre-heat the BB and go quick.
Do not hacksaw off the axle ends - that will not help.
Once you have one cup off, the axle will slide out that side along with all the bearing balls. At this point, it is possible to hacksaw through the remaining cup providing a space where it can compress and release the thread. It is hard work and easy to cut into the softer frame when you get near the end. I've done this once.
Update: I'd suggest you start by fitting your park tool in place, and then find some washers that will fit over the axle to raise the level, and then use some sort of clamp on the flat part of the axle where the cotter pin goes. Ideally it should be kinda wedged together so you can tap on the side of the tool with a steel hammer.