The fact that your vehicle was uninsured would not bar you from making a claim under the other driver's policy for your damages. You need to get the hit-and-run driver's insurance information and file a claim. Since you have video of the incident and witnesses who got the license plate, you shouldn't have too much trouble persuading the insurance company that their insured was responsible.
This doesn't apply to your case since you weren't injured, but as a point of interest: a Texas legislator tried to get a bill passed that would prevent uninsured drivers from making a claim for anything other than the amount of their medical bills if they were injured in an accident that was not their fault. The thinking was it would encourage people to carry insurance.
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Simply submit the claim to the other driver's insurance company.
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Since it happened in a parking lot, you cannot get a ticket, but otherwise you may have gotten a ticket for no insurance even though it was not your fault. If you ever are in a collision that is your fault and you have no insurance, you can have your license suspended.
This is not legal advice. You should always discuss the specifics of your issue in person with an attorney. Be aware that there are time limits on all claims that depend on the kind of claim, so do not delay in seeking an attorney.
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