Office Neighbor Yelling At Me For Walking In Parking Lot

For context, our office is centralized, theres a sheriff’s tax office, justice of the peace, dentists’ office, and a couple law offices, and then where I work (legal-adjacent firm).

So all of these offices are located in the center, with a middle roadway splitting up half the offices, and surrounding the circumference of our property is a wonderful parking lot to walk during my breaks.

There are no sidewalks on the outer edge, and to get my walks in during my breaks, I would be u-turning every 3 minutes to walk on the very small amount of sidewalk space that we have. I have been walking on the outer edge of the parking lot, near cars (without ANY issues - always yielding to parking and departing cars and gaining smiles in the process), and I was assuming this was a public space.

The dentist just came and requested me to not walk on the parking lot in front of his office, claiming he doesn’t have insurance and has lots of traffic coming in and out (I have literally seen 3 cars actively park there while on my walks) and so I’m not covered if hurt on “his” parking lot.

Apparently, other coworkers of mine have gone through this. Our building’s manager is “working it out,” yet this has been ongoing for a couple months, as I’ve come to learn.

I just want my darned walks. I’m injured on my wrist so I can’t lift weights rn, I’m a sedentary office worker, and walking on my breaks was helping me SO much like omg.

I’m in Location: Austin, Texas. Is a parking lot spanning multiple business offices shared, or does he own (and therefore responsible for potential civic duty breaches) parking lot space? I don’t have access to the lease, and was informed to comply in the interim while our building manager works it out. There are truly no alternatives to my walks besides walking the same 500 feet of sidewalk back and forth. Which, to be quite frank.. sucks. :(

Edit: either way I know this is out of my hands. The last thing I want is to cause issues, and I want to keep my job. But, I’m just curious. I wouldn’t be able to even do anything with any advice, it’s above my paygrade. This is mostly a rant, in all honesty. But I’m curious about what civil law says here.