How old will one have to be if a car accident occurs with out any type of car insurance, 1 DWI and 1 ticket and if one gets sued by other individual?

I'm not clear as to your situation.  What do you mean by "how old does one have to be"?  Are you under 18 and are worried that your parents will be sued?

There can be claims involving a minor's use of an automobile (including motorcycles) which harms another.  The common tort for a parent's civil liability is known as a claim for negligent parental entrustment of a motor vehicle.  The basis for such a claim is two-fold.  First, the parent must have exercised some control over the vehicle; second, the parent must know of a child's tendency to drive recklessly.

The fact that a parent owns a vehicle is not enough to qualify under the claim of negligent parental entrustment; in fact some courts have held that the parent must be physically present.  In addition, with respect to a parent's knowledge of a child's propensity to drive recklessly, some courts have ruled that a parent's knowledge of prior traffic accidents may not qualify as knowledge. 

However, all this does not mean that such a claim cannot successfully be made.  Because these type cases are very fact specific, and the law is always changing, your family will need to consult with a Texas personal injury attorney.

Also, parents might be held liable for alcohol consumption by a minor child which causes injury to others. 

Again, speak with an attorney in your area.