I thought that claim by the parks department person sounded a little hard to believe. I had previously heard that the best accuracy was about one centimeter but it looks like it's a bit better than that.
Food for thought:
After I saw the accuracy claim I called the surveyor I work with that surveys our deep water well locations (think surveying a spot 10,000' under water) subway tunnels, and surveys in the GPS systems on ships (ie he is one of the best surveyors in the world). He told me 2 things. The first is the most accurate survey possible is about 5 mm by waypoint averaging in 30 min increments over the course of the day to grab different satellite constellations and then performing post processing using tie in points for reference that have been surveyed multiple times over the years. The second is that there really is no "survey grade" GPS. Without the post processing our handhelds pulling off GLONASS, GPS, WAZ, etc are just as accurate if left to waypoint average for the same length of time. Without post processing he says the best accuracy to expect is ~ 3 ft. I confirmed this as he surveyed our farm for us with me clearing survey lines. At the GPS points he waypoint averaged at for 30 minutes and then post processed I can match within 3 to 6 ft if I waypoint average for 3 minutes.
I emailed the San Antonio Parks Dept this morning and got this quick reply this afternoon.
This is state plane coordinats for the more expensive Survey grade GPS, not your less than hand helds.
The accuracy of the gps numbers are to the 1000 of an decimal inch, they geocachers would have to calculate it to what they use.
Coordinates for this spot for Geocaching is 29* 29’54.91” N
This beta conversion tool gets you pretty close.
Received another reply to say the conversion can be down via GoogleEarth.
I found another conversion tool but you have to enter the zone number which appears to be 4204. https://gis.tulaliptribes-nsn.gov/Home/FrequentlyAskedQuestions.aspx
More on the History of State Plane Coordinates. Used in the US.