Easy to find car park areas with good public transport into the city are the airport areas.
All of them have the official as well as alternative parking areas.
Heathrow is on your way to London, Gatwick on your way around towards Dover, Luton and Stanstead can be useful if you plan to use the ferries at Harwich.
You can also look in booking a hotel out of the city for one night before and one night after the visit to the city and contact them ahead of time to talk about the parking 'problem'.
I stayed in Maidenhead recently and while parking at the hotel was full for the weekend, during the week they did not need all the spaces.
So will a lot of hotels near London and alternatively, a lot of the hotel owners will know alternative parking spots near their hotel, they have to because their guests often need them. By the way, Maidenhead will not be in the best location.
And of course, all of the commuter area will have good transport into the city and most of the train stations will have some kind of parking at the station. Most of those are not free however and might have a limit on the time your car may stay.
If you are on the far end of the commuter beld, at at a small station without direct trains, you are more likely to find free parking at or near the station.
There are too many options to give you names of towns and villages.