If you park in Twickenham, it will cost about £8 for parking in the car park across from the station but there are reasonable beer prices at Wetherspoons - around £3 a pint.
You can also stay at a meter for about £1.20 an hour which isn't too bad but there's normally a limit of two hours. If you are quick on the way out and in this might just about work. You'd probably not be a drinker and unusual to run in and out!
If you park at the Twickenham Stoop, it will cost a more reasonable £5 but the beer is more like £4 a pint so it might be worth doing a bit of short stay and then driving in.
There's not any free supermarket parking to the best of my knowledge, Tesco ain't a million miles away but they are greedy for cash.
If you park in the street the the council are red hot for fines and charges and virtually everything is considered to be out-of-bounds/residents parking.
If you want to park for free I would drive onto the private land of a business which is shut for the weekend, maybe a small premises for a Doctors or Accountants, but without a clamping notice.
You can just ignore any fines from private car parking firms - I always do.
It sounds a bit petty, but when they want £25 for parking on a Union matchday, then this ain't going to come from me that I can tell you!
Rammo