Have you tried Kiwanis Park on Market St/Main St?
I'd send you a link, but I can't find any website for it. This was about as close as I could get on Google Maps:
Link
It's near the VA between Market and Main Streets and between Sydney St and Pactolas St.
It has been at least 8 years since I've been there, so maybe things have changed since then, and there's a new spot in Johnson City for pick-up basketball, but that was the place to be in the '80s and '90s, and I think I went there in the early part of this decade as well a few times.