直接向對方保險公司索賠

get a police report and witness if any.
it should state that your car was parking in the lot.
since the other party admitted it was his/her fault and he/she already reported the accident to his/her insurance company, his/her insurance company usually will contact you and send someone to inspect your car for damage determination (or you may call them), then they will write a check to you let you fix by youself or let you fix your car at their assigned repair shop who will directly charge to the insurance company. you may bring your car to get the damage estimate. to do so you will know how much it will cost. however, you need to be aware that the estimate always be much higher if you said you will pay for the cost than that the service fee (including metrials, albor, and tax) they provide to insurance company. since you have the comprehensive coverage, your insurance may say that they can take the case and pay for the cost then get reimbursement from the other party's insurance company. someone advice not to do so since your company may raise your insurance premium later. do it yourself. good luck.