I agree with what Waywardangler added to my “fix-it” directions. And actually, if I were fixing it I would use a hardwood dowel pin instead of the “allthread” because a dowel pin will “flex” a wee bit if pressure was put on the joint when one was using the parasol, aka: a gust of wind. But, replacing the “allthread” with a dowel pin requires a bit more expertise to insure that the handle is straight when the epoxy hardens. And you only get “one shot” at it and if it’s wrong, you got to drill it out and do it again.
If the “allthread” wiggles, it should un-screw. And I can’t imagine it being glued in there. If the holes were drilled slightly “undersized” then the “allthread” would cut its own threads into the ivory when screwed into the holes. Whoever repaired it first, knew what they were doing because …….
NO WAY could an amateur drill those “matching holes” in the two (2) handle pieces. I’m fairly sure that it was done by a machinist using a lathe or a boring machine.
So, the “wiggly” end of the handle should un-screw and if the “allthread” is stubborn at coming out of the other piece then one can heat up the end with the nuts on it with a propane torch or whatever and then use the wrench to twist it out. Just wrap the handle with several layers of cloth and secure it “snuggly” in a vice before doing so.
Tales, take it to a Machine Shop and they will re-fix for you. Have them put a nylon (or whatever) dowel pin back in it and it will then be stronger than the original.