Yes, if you are not joining a guided tour and driving to the eruption site on your own, the parking fee is 1000 ISK (≈ 8 USD) and can be paid electronically at Parka.is. The payment is valid for 24 hours. There are signs at the parking lots with instructions.
There are currently two parking lots that you can choose from, Parking P1 or Parking P2. You will need to pick the correct payment site accordingly:
Parking P1: https://www.parka.is/pay/geldingadalir/
Parking P2 (Stóri-Leirdalur): https://www.parka.is/pay/volcanoskali/
According to parka.is, the payment is necessary to pay for the infrastructure on-site and maintenance of it. The intention is to make access better and increase people's safety. The parking lots are monitored with cameras and if parking is unpaid, an additional fee is added and a bank claim is created. The claim usually goes out one day later, so if you have trouble paying on-site, you are able to finish the payment when you arrive back at your hotel.