The 'county town' is Durham which is a city. The largest towns are Darlington, Hartlepool and Stockton-on-Tees.

Durham County Council

The website states
'Motorhome parking: motorhomes are permitted in all of our car parks, however, cooking and overnight sleeping is not permitted.

There are 3 park and ride services into Durham on 3 sides of the city.
Sniperley DH1 5RA W of city - 400 m north of the A167/A690 jct. No size or height information.
Howlands DH1 3TQ S of city - On the A177 opp golf course. No size or height information.
Belmont DH1 1SR E of city - on A690 300 m east of Jct 62 of A1(M). No size or height information.

Other car parks include:-
Durham - Sidegate DH1 5SY - 107 spaces. No height information.
Durham - The Sands DH1 1SW - 136 spaces. No height information.

Weardale - OS maps OL31 North Pennines and Landranger 92 Barnard Castle and Richmond..

There is room to park motorhomes in the following Weardale villages:

Westgate: Parking opposite the caravan/camping site on A689 which is the main road that runs through Weardale. Access is excellent.

St. John’s Chapel: Parking on the car park near the Auction Mart immediately after the St.John’s Chapel sign as you enter the village from the east. Also on the A689. Access is excellent. MS

Darlington Borough Council

Buy two tickets if you can't fit into a bay. Display both tickets.

Most Darlington car parks have a height restriction. The following do not:-
Central House, Gladstone Street - 46 spaces.
Chestnut Street, Chestnut Street - 109 spaces. Also a lorry park.
Hird Street, Hird Street - 13 spaces.
Park Lane, Park Lane - 83 spaces.
St Hilda's, Tannery Yard - 15 spaces.
Park Place East and West has 210 spaces. The height barrier is set at 9Ft 6inches.

Hartlepool Borough Council

No information for motorhomes. Following an email: ' There are dedicated parking bays for long vehicles at Seaview cp at Seaton Carew. It is locked at dusk.'

Hartlepool - Transport Interchange 1, 2, 3, 4, 5 - Lynn Street TS24 7DS - 138 spaces but no height information.

Stockton-on-Tees Borough Council - Stockton and Yarm.

There is no information for motorhomes on the website.

The above link takes you to the car park locations in Stockton and Yarm. There is no information about size or height restrictions. It does show long or short stay.