Our neighbor has a holiday apartment that is used only one month a year and the rest of the time it is vacant. The neighbor has an open parking which is also vacant unless the neighbor is using the apartment. Does one need permission to use the parking on an ad hoc basis?

Shalom If it is not being used and it is reasonable to assume that the neighbor would allow it, you can use the parking space. The neighbor can, of course, use it whenever they want, even if it is extremely inconvenient to you. Kol tuv