Can a insurance company file paperwork to suspend my divers license for a fender bender in a parking lot?

In your state, if you drove a car without motor vehicle insurance and if your state has a law that allows your license to be suspended for this act, then yes, your insurance company can certainly file a request for this and whether you are entitled to a hearing or not or just get an automatic suspension will depend on your state's laws. Contact the department of motor vehicles and find out if there is information on this. As to your motor vehicle, if you attempted to put a claim in with your insurance company because your car was hit or the woman who hit your car attempted to file a claim or told her insurance company, then the issue becomes whether you were also at fault. If your car was parked and you were parked legally then this should have nothing to do with your insurance company. If you were also driving and you should not have been driving because you did not have motor vehicle insurance, but the accident was 100% not your fault (police report says it was the person's fault for example), then again it should have nothing to do with your insurance company. Review the facts and see what you need to do. Your other choice is to file a complaint with your state's insurance department that regulates motor vehicle insurance and see what it can do to help.