Yes, your parents will need to notify their insurance company about your licensing status and typically an insurer will then require them to add you to the policy as a driver. It does not matter that you do not own any of the cars in the household just that you have access to them and will likely drive them and thus are considered a risk factor that an insurance company will want to use for rating purposes.
The Texas Department of Insurance (DOI) Auto Insurance Guide backs up this idea. Here it says young drivers must comply with the state's financial responsibility laws, just as older drivers do. Most young drivers, however, have the option of satisfying their legal requirements by being added to their parents' auto policy. Adding a young driver to a parents' policy can be expensive, but it is cheaper than taking out a separate auto policy.
A parents' policy covers children living at home or away at school, even when not named on the policy. Even though children are automatically covered on their parents' policy, it is important that they be listed on the policy as soon as they reach driving age. Insurance companies are required to charge the correct rate, based on the classifications of the drivers in your family. If you do not have all of the drivers in your family listed on your policy and the company learns about them later - because of an accident claim, for instance - the company will bill you for the extra premium you should have paid.
The TX DOI also notes that you can sometimes remove a teen-aged driver from your policy by purchasing a non-owner policy. However, this usually is a bad idea. A non-owner policy merely provides additional liability insurance when driving a non-owned vehicle.
If you have accident while driving your parent's car neither your parent's policy nor the non-owner policy would pay for the vehicle's damage. Your parents might also be unprotected financially if held liable for an accident caused by you, a minor child. Finally, if the non-owner policy is rated properly, your liability insurance might cost as much as or more than if your parents added you to their policy.
Rates are usually high for newly licensed drivers since they are inexperience and statistically shown to be risky drivers. Male drivers under 25 and unmarried women under 21 have the highest rates usually.
The costs of adding a teenager will vary depending upon your parent's insurance rates already, your geographical location and all household members' driving records. For instance if you live in a congested area where your are more likely to be involved in an accident compare to a more suburban or rural area the rate to be added to your parent's policy will likely be more.
In very general terms adding a teenager may cost $1200 to even $2500 a year. On average, a 16 year old can expect to pay 2.52 times the base rate for their insurance. So if a parent of a teen now pay $500 for a year of car insurance, then adding a 16 year old newly licensed driver onto their policy could raise their rates to (again on average) $1260 for a year of car insurance.
There are discounts that your parents can ask about that may help bring down their rate increase a bit for adding you to the policy. Discounts for defensive driving and driver education courses for young drivers and a student with good grades are a couple that your parents can ask their agent about.