You may be able to. File a claim for the property damage with their insurance company. if no good, try a small claims court case. You may be barred from any persoanl injury claims. Good luck.
The above is general information only and is not legal advice. The information provided does not form an attorney-client relationship, and should not be relied upon to take or refrain from taking any action. I am not your attorney until we sign a retainer agreement.
Because you have no insurance, and if the owner of the other car had insurance, you are limited at best, to a mini-tort action with a maximum recovery of $500 or $1000 depending on when the incident occurred.
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You will likely be completely barred. If you are uninsured, you cannot file a mini tort claim to recover the first $1,000 of vehicle damage and you must pay for all of your own car damage, even when it is another person's fault and you are completely innocent. Why? In Michigan, if you are driving your own uninsured car or truck and it is damaged in a car accident, or even if someone bumps into your car in a parking lot as you wrote, then you are sadly responsible for paying for all of the associated repair costs.
The Mini Tort recovery for uninsured drivers stopped on October 1, 2012, when uninsured drivers were disqualified from recovering under the Mini Tort law.
Based on amendments to the statute in June 2012, which are contained in Public Act 158 of 2012, “[d]amages shall not be assessed if the damaged motor vehicle was being operated at the time of the damage without the security required by section 3101 [i.e., No Fault auto insurance].” (MCL 500.3135(4)(e))
Here is a blog on this topic that may be helpful for you:
A roundup of the best tips and legal advice.