The cost would be the same for the expedite process, whether you go to the USCIS branch for emergency or expedite online. You need to show emergency documents. It has to be either humanitarian related OR family emergency. I do not know about humanitarian, but for a family emergency, you need to show that your family member is in critical condition. You will need a document to verify this. It is better to have a letter from the hospital ( with hospital letterhead) briefing about diagnosis and prognosis. Then you need to verify that he/she is actually your family member. Air ticket may or may not be required. I had one, but they did not even look at it. Their application will ask you the expected leave and entry date. The parole will be valid for only these days ( not for strict 1 month).
In my case, I had a relationship verification certificate (original ) with photo and name matching the letter from the hospital. The hospital letter had a brief deskription of the diagnosis/prognosis. I had my plane ticket to depart the next day ( I was going home regardless of approval). If you have time, do not do this. They were very doubtful that they could approve the same day and were expecting me to come the next day which was not possible in my case. I was lucky that I got approved the same day. I filled the expected departure ( flight date) and re-entry date which was of the duration of 35 days, and that is what I got parole for ( single entry ). I did have x-rays and other scan reports with medical preskriptions and purchase bills submitted. So try to get as much documentation as possible.