I used to park at Arnos Grove station regularly on trips at one time, as there was usually space and I knew the area. I haven't been back there for years though, so I have no idea what the situation is like now. Limes Avenue where my grandparents used to live used to offer parking spaces (and is only 2 minutes walk from the station), but even 15 years ago it was starting to get a little crowded, so I have no idea whether you'd be able to find a space there these days. You might need to go a little farther away -- The far end of Betstyle Road or one of the other residential streets, or perhaps the far side of the park).
Oakwood station: I know from an old family friend that the local council imposed a lot of parking restrictions in the residential side streets there a few years ago specifically to stop commuters parking in them for the whole day. I'd forget Southgate: Even 15 years ago it was also impossible to find anywhere to park within a reasonable distance.
You might also look at parking around the end of the northern line. Totteridge & Whetstone station has (or at least used to have!) a small parking lot, and High Barnet (end of the line) has parking too. No idea about current time limits or charges though.