Someone hit my parked car. Next steps or advice?

You've done everything you need to do.  Any questions you have should be addressed to your insurance company, not Y/A.   our insurance company will handle everything.  That is why you pay them.

Some of this makes no sense.  You are the owner but it's your mom's car?  That's actually illegal.  You can't insure what you don't own.  She can't insure YOUR car.  

You don't "make a claim with U-Haul."  You report the accident to your/your mother's insurance carrier, and THEY handle the claim.  Why are you  pursuing the other party's insurance when that is the job of YOUR/your mother's insurance carrier?

What should you do?  Butt out before you screw the claim up so badly that no one pays.

Where are you that the Police won't investigate an accident on private property.  That is NOT the case in my State.

Why are you claiming against U-Haul instead of the driver of the U-Haul vehicle?

I'm an insurance adjuster.  This question makes me cringe.  I cannot do my job for you when you are interfering with the claim.

Since you have a new vehicle you have "Collision coverage" on it.   Use your own insurance and let your insurance fix your car.    Waiting for U-Haul is a big mistake.   Do not let U-Haul fix your car.   Do through your auto insurance company for the repairs.    That is what you pay them for.   You don't need to do anything.

Call your insurance company.

You  DON'T PURSUE the other person's insurance.  You contact yours and they do the pursuing. That is why you pay them the big bucks. You insurance company WORKS FOR YOU. The other works for the other guy.  You talk to YOUR AGENT and he seeks compensation.(because your insurance does not want to pay for the damage out of their bank account SO THEY CHASE AFTER THE OTHER INSURANCE TO COMPENSATE YOU.
. Police won't come because there is no injury or death. It is simple property damage.
.  That is the way it has always worked.
You will find this information in your Driver's Training Manual too.

You've given all the info you have to your insurance company, you've got all the names and vehicles involved, and they'll handle it from here.  You've covered all your bases.

A person who hits a lawfully parked vehicle is 100% at fault.

You need to get at least 2 estimates for the cost to repair this. Submit these to Uhaul, also keep your insurance company in the loop, if Uhaul drags their feet or refuses to pay you might be able to have your insurance company pay to have it fixed and they can go after Uhaul's insurance. If not you can file a claim in small claims court. Make sure you have lots of pictures of the damage.

If it's your mom's car, how can you be the legal owner?  If her name is on the Title, SHE is the legal owner and the insurance is probably in her name too.  
 
YOU (or your mom) don't pursue the other driver's insurance company and neither of you files a claim with U-Haul. Don't talk with the driver's insurance company or with U-Haul at all. Do exactly what you've already done: Contact your own insurance company, and your insurance company will handle all the pursuing and claiming and everything else for you, and you just sit back and let them do it. You obtained the driver's insurance information, so give that information to your insurance company - that's all your company needs. 

Find out if you have rental vehicle coverage or get UHaul to agree to pay for a rental.