auto accident, personal injury lawsuit under TX tort laws

You have asked a number of questions.
First even though the light was yellow as you entered the intersection, it turned red while you were in the intersection. You are responsible for her damages. Further a yellow light is not supposed to encourage you to continue driving. Rather it means caution, and it is wise to stop unless it turns yellow as you are passing through the intersection. In other words, you should rarely go through an intersection once the light turns yellow to your side of the traffic.

Second: A payment plan may be acceptable but you will owe interest. You should check the rate of interest. Further if you fail to pay her, she can lien any property you have or obtain. She can also take a part of your pay check (usually 10%) plus the interest. It might be wise to get a second (or first job) even as a full time student.

Good luck.

You have done pretty much everything wrong so far, from taking a car without consent (that's stealing), to running a red light (it changed to red while you were still in the intersection).

If you have your own auto insurance, that company might help you. The insurance on the car has no duty to help because you did not have permission to drive the car. There is a slim possibility that you could show implied consent if the vehicle owner routinely allowed you to use the car without express consent. I would at least try this argument.

There is also a chance you could win an argument that the other driver did not allow the intersection to clear before proceeding through. Just because you have a green light doesn't give you the right to run into someone who is already in the middle of the intersection. You do have a duty to try to avoid a collision even if you have the right of way.The fact that the impact was at the rear or your parents' car is in your favor in making this argument.

As for a payment plan, that would simply be up to the other driver. Your argument there is that you have no money to pay now, so taking a little at a time is better than getting a judgment that you won't be able to pay.

I wish you all the best, but you're not in a good position here.