A driver under the influence hit my car while it was parked on the street in front of my house. Should I file a claim directly with this driver's insurance company? I reported it to my insurance company, and they said because I didn't opt for towing and rental car coverage that I will have to pay out of pocket for those costs. I think that the other driver's insurance company should pay these costs since the accident was his fault. What do you advise?
Yes, you should contact the other party's insurance coverage. You should explain to the claim adjuster your situation. Based on what you have stated, it appears as though you should be entitled to a rental car. It is also possible that if you incur these expenses out of pocket that you could be reimbursed for them by the other party's insurance company.
If you do not want to incur the cost of a rental car and then go after the other party separately for that expense than you should put the claim through the at-fault party's property damage liability coverage. If you do this instead of your own Collision coverage there would not be a deductible to be paid and a rental car should be provided to you while your car is being repaired. If you want less out of pocket expenses (that may or may not be paid back to you) than going through the at-fault party's insurance to make a claim instead of your own would be the way to go.
