There is an RV park on the South side, on the north bound access road to I-35, Austin Lone Star RV Resort. Also, try Lake Travis RV park on FM 620, northwest. If you head out to Leander/Liberty Hill, there are several in that area that I think will take an older rig. None of these is convenient to the City but the only one that is, is downtown and always full. Always.
There is one that may fit your needs between Parmer and US 183, on McNeil. I don't know the name of it and have never stayed there but have driven by it lots of times, lots of trees and shade! It's on the north side of McNeil, easy to pass and not notice.
Good luck and best wishes.