I am planning on hiking a loop of mostly the AT. Where is the best place to park at the state park and what is the process of paying for a one-night backpacking permit/parking?

Park in the backpackers lot near the Massie Gap parking area, and pay for the pass at the check in station at the main entrance off US 58.

We have a backpacker parking lot. You pay at the contact station or, if no one is in the station and the office is closed, there is a self-pay envelope and drop available. It's $8 per night. Here is some additional guidance: https://www.dcr.virginia.gov/state-parks/grayson-highlands#recreation