My question involves a traffic accident in the State of: Texas

I was traveling down a neighborhood street for one lane of traffic in each direction, divided by a yellow line. The side of the street I was traveling is designated a no parking, tow-away zone while parking is allowed on the other side (in the opposite direction I was traveling). I approached an occupied car that was parallel parked against the curb on my side of the street, so the car was illegally parked in a no parking zone. I saw the parked car and noticed no turn signal to indicate that it was preparing to pull out.

Since there's not enough space for two-way traffic if a car is parked on my side in the no-parking zone, I veered left straddling the yellow divider to pass the parked car. When I was finishing my pass, I veered back right to be fully in the lane. The front driver side tip of the other car and the passenger side front door of my car made contact. I stopped and exited my car to investigate, and the other car (which was originally parked) had started to pull out forward and slightly left some time while I was passing it. The other car's wheels were turned left into my car.

The space I was veering right back into would have been clear had the parked car not started to pull out. I noticed where the front of the parked car was when I began to pass so I'd know when to start veering back to the right. However, at some point while my attention was focused forward at the coming intersection (just about 50-100 feet away), the parked car began to pull out. At the scene, the other driver said that she was pulling out.

The other driver claims to have saw me after she started pulling out, so she stopped before I hit her (as she claims). It seems like a clear case of her failing to yield right-of-way to me, a passing car. However, when I called her the next day, she said that she believes we're both equally at fault. Now I have my insurance company telling me that she filed a claim against me saying that I'm at fault. After dealing with her insurance company, GEICO, their adjuster called me and said that we're equally liable. I disagreed with GEICO's adjuster, who told me that they won't find their own customer at fault, and only my insurance company could do that.

Who is at fault here, and is it right for GEICO not to find fault with someone because they're a GEICO customer?