Why do you need off street parking? It does limit you hugely. The nicest parts of London are often Victorian/Georgian terraces and they don't often have it!
Most areas of London are controlled parking zones - ie residents only. Some areas will have issues but I struggle to think of any examples in my life where not being able to park on your street is an issue.
As you don't drive, I would say don't just consider the public transport for commuting to London/travelling in for a day trip. Your day to day life is much more about being able to walk or cycle to your local facilities.
The more small town/rural/villagey options suggested here may be ok for a commute into Waterloo/Kings Cross but may be a real pain for going to the library, swimming pool, hair salon, DIY shop etc but realistically almost everyone there will drive and local public transport will likely be very poor.
I would suggest looking at Waltham Forest borough - they've done a lot of work to make it a more pedestrian and cycle friendly area (they call the scheme Mini Holland) and part of that includes the concept of a 15min area, where you can reach without a car almost all your needs in 15 mins
It is London so yes there will be some grotty blocks here and there. Some parts (walthamstow village) might be a touch too pricey for you and it's Victorian terraces so you would need to compromise on off street parking
Palmer's Green may be of interest but while the commute into town might be straight forward, it is the sort of area where lots of people drive locally.
Turnpike Lane - some people will be horrified. But the Harringay Ladder and Gardens as firmly up and come already and is out of your price range now if you want a house. Also - again - no off-street parking.
Green Lanes has a lot of Turkish restaurants (and they are excellent) but has recently diversified and there is a bit more variety going on including more hipster cafes/bars etc. Also has fabulous Turkish greengrocers, butchers and bakers - food shopping opportunities that would be the envy of many.
Have various friends around Chestnuts and Downhills Park in lovely Victorian houses and kids happy at local primary schools. Easy to function without a car but again - most will not have off street parking as they are your classic terrace.
Yes, there are grimmer parts of Haringey but really, not everyone is out getting shot or stabbed on a weekly basis. It's actually frightfully middle-class in places.
Wood Green shopping centre is quite grim, can't deny that.