Typically, an insurer would prefer to handle any and all damage incurred as to avoid any unnecessary future risks of property damage or bodily injury caused by the damaged structure. This is a claims-related question that would be uniquely handled by the claims department of your insurance provider.
If you were listed as a driver under your parents’ policy, you would be considered covered by insurance for however long you were on their policy. Be sure to shop around for your new policy before your parents remove you from theirs to avoid a lapse in coverage.
It is best to wait and confirm whether the at-fault party has insurance or not. If they do, the insurance adjuster can take a look at your vehicle and collect the necessary data to move forward in the claim process.
Since you were driving your company’s vehicle, your personal auto policy would not apply. A company vehicle like a backhoe or other construction vehicle should be covered under the company’s commercial auto policy.
It's possible that you may be incurring a surcharge for the out-of-state vehicle, but I don't know any company that currently employs this practice. However, the vehicle history you mentioned can change your rates.
If you have assets you are trying to protect and have maxed out your liability limits, I would recommend getting an umbrella policy. They tend to be very inexpensive and will offer more robust coverage.