The Berwick Inn (pub) offers parking facilities. You can find it through BritStop. If you request politely, you may be allowed to stay overnight and have dinner at the inn. Breakfast is also available.
I stayed at a nice pub with my friends last night. The food was good and the pub had a pleasant atmosphere. The only downside was that the noise of passing trains could be heard, but it seemed to stop after 10pm and didn't start again until 7am, so it didn't disturb our sleep. Additionally, the pub had a large and convenient parking area.
I was informed in advance that the pub and restaurant were closed on Mondays, but I was allowed to stay. There was some noise from trains, but it was quiet at night. I would recommend this place and plan to stay again, but I'll make sure to visit when the pub is open as it seemed great. The staff members are very polite and friendly.