There are parking restrictions in place around the ground on match days, and non-residents that try to park in a restricted zone will find their car hit with a Penalty Charge Notice or even towed away. The best thing to do is to move out of the restricted area to somewhere that's still on the train or tube network and come in from there, but there are probably other private parking options a bit nearer to the ground that might serve better.
The new Spurs stadium is, in fact, now designated a 'Public Transport Destination,' meaning they are doing as much as they can to discourage car travel. It is claimed that over 60% of fans went to the match using a car in 2003, dropping to 42% in 2014. They aim to get down to a maximum of 24%.