We don't have "automobiles" round 'ere. Just cars or, if you want to be real posh, you can use the term "mechanically propelled vehicle". With that important clarification out of the way, I see no reason for you to be driving around London in the weekend. It's likely you can find free on-street parking on Sunday, however Saturday is much more difficult- I'm willing to bet there is absolutely nothing to be had for free in zones 1-3. But, even if you did find somewhere to park the bloody thing, the traffic will still be horrendous and generally moving about by car will take you a lot more time and effort than using public transport.
So just do a rental starting Monday morning, and, if going North, be warned that the first half-interesting city you'll come across (that'd be Northampton) is a good 70 miles away, with the really interesting options (Leicester, Derby, Nottingham, Sheffield) even further North.