Where can I park in Bath for free, so that it's convenient for the town centre? I will be there 22nd Dec - 29th Dec, so on both holidays and normal weekdays.
Short answer: you can't, sorry :/ Parking is at a premium in the tight and narrow streets of Bath.
Long answer comes from Wikitravel:
It is very easy to get lost in Bath, as a lot of it is one-way and there's a traffic system that prevents you driving from one side of the city to the other. You have to go out on an unofficial ring road and re-enter the city. Furthermore, the high population density, the lack of a city bypass and the low capacity of the old narrow streets means that congestion is often horrendous. In particular, on Saturdays the car parks will all be full, and the roads will be blocked by people queueing to get into these car parks, a problem made worse since the opening of the new Southgate car park. At peak times, it can be quicker to walk from the edge of Bath to town, rather than driving and finding somewhere to park. The short answer - don't drive in Bath.
Parking in central Bath is often a nightmare and two hour limits apply on many streets. Most of the smaller long stay car parks will be full by 8:30AM during the working week so you have to get in early. Major central multi-storey car parks are based at Walcot Street, Manvers Street (near the train stations) and Charlotte Street (off Queens Square). Average 2010 rates are around £3 an hour - or the more prohibitive pay and display in central bath at £1.30p per 30 minutes in the most convenient street locations. Many parking bays are "residents parking only" so check before leaving your car. Traffic wardens are very efficient so don't even think of parking on a yellow line or going over your time limit. On Sundays and between 7PM and 8AM other days most parking is free, however check machines for exact details.
The best way to drive into town is to use the park and ride facilities  when travelling into Bath for the day. You can park for free and then take a bus for £2.20 per adult return (round-trip, discounts exist) right into the city. The only downside to this is that the last bus leaves at 8:30PM, so you can't use this service if you're staying in Bath late.
I know the date has passed - but for future reference you might consider 2 options -- not free - but maybe cheaper than council or NCP:
ParkatmyHouse - from the site: created to connect home and business owners who would like to earn money from renting their space with drivers in need of a convenient, safe and cost-effective place to park.
Not just Bath, but the whole UK! Particularly popular places include Twickenham, Surrey/London for the Rugby - and I'd imagine those with parking areas in Bath would be popular too for the same reason.
Raileasy - If there is any possibility that a car is not needed - this site is aimed at travel to various events that have rail stations nearby.
(I have no involvement in either service.)