If it's a half-way decent school in a place like That London where everywhere is hideously oversubscribed and house prices are ridiculous, it's quite unlikely that you're the only person with an education who's moved into the area.
I suspect that PP are right and you've only seen the non-waged parents (the others have managed to get their first Proper Look at the school on a different day to you). I may be wrong but I suspect in London non-waged parents fall into two distinct groups only one of which is represented at your school 
My boys went to a primary school which had two definite groups of parents - the ones who were, or had been, associated with the university and the Long-term Locals, some of whom were not particularly pleasant characters. Neither child had any social problems caused by the other kids, and I did spot the parents I had most in common with.
Spent most days for a few years desultorily chatting to one of the Locals - though she wasn't an actual local as she'd spent her childhood in the county to the north, and she was considered The Posh One in her family; and to another Local who also wasn't a local because he came from Ireland, and was actually the child's grandfather (he's dead now, have a quaff of nectar on me Paul); and to an Iraqi woman whose husband was doing a post-grad course at the university, who spoke English marginally, and whose child arrived at the school in Y1 speaking no English at all and ended up being S2's best friend for four years. Oh, and one of the childminders. Haven't spoken to any of them since, except one meeting with Not-Local Woman in Tesco a few years ago.
I did also chat occasionally to the parents of S1's friends, but most of them worked. The ones I saw most were both nurses (ie shifts). Saw them occasionally since primary school ended - one couple was at secondary school concerts, one turned up to interviews that I was part of as a school governor, the nurse I have met walking to town, one or two in Tesco.
In fact the parent I've probably seen most since the boys left that school - apart from the fellow-governor, whose kids were older - is one of the Locals who has six children (spread over 20 years but I think all the same father) and is now a cleaner at the school.
Still feel a bit guilty about playdates (they can be counted on my fingers) but there you go. Parties weren't a problem. Boys have organised their own social life - S1 was having mixed sleepovers in Y11 and S2 has a girlfriend and a core group of friends who play D&D. I haven't really spoken much to any of those parents even though we have a lot in common!
On that note, a plea: if you do see a parent who looks a bit educated and out of place, could you make an effort? Some of us are painfully shy...