There are 4 public parking lots when you drive to Neuschwanstein Castle. This map shows where they are (the red highlighted areas; click on them for their official names):
All these parking lots are in the little village below the castle (the village is called Hohenschwangau); this village is where all the tourist stuff are located, like the castle's ticket centre, bunch of restaurants restaurants, hotels, etc.
Keep in mind that Hohenschwangau village is the only place where visitors can park their cars. You cannot drive up to the castle and park there. You must park in the village in one of the four public lots or private hotel lots if you're a guest there
In fact, when using GPS to navigate to Neuschwanstein, put in "Hohenschwangau, Germany" instead of "Neuschwanstein Castle" to make sure that your GPS doesn't inadvertently take you up to the castle
From the parking lots in Hohenschwangau village, it's about a 25-40 minute hike up to the castle. There are also buses and horse-drawn carriages you can take to get up to the castle as well
Parking at Neuschwanstein Castle costs €7 per day flat
Payment is cash euros only, so be sure to bring some cash with you. When you drive into the lot, you will pay the attendant the amount upfront. The attendant will give you a slip to put on your dashboard