You could look at Telegraph Hill (which is the conservation area in SE14) and Brockley (just down the road). If you live near enough to get a place at Haberdashers' Aske's Temple Grove (primary section of an all-ages academy) your daughter will be guaranteed a place at the senior school, which is one of the most oversubscribed comprehensive schools in Inner London. Telegraph Hill Park is lovely and has a playground.
Brockley has a very good girls' secondary school - Prendergast - and several good primary schools. You'd need to make sure you lived very near Prendergast to get in there - half a mile? The secondary schools in Lewisham are generally improving, I'd say, and some of them are very good indeed. Hilly Fields, Blythe Hill and Ladywell Fields are all lovely parks with very good playgrounds. A little further out but still very close to New Cross (direct train line and probably direct bus routes too) you have Forest Hill, which has both a good park and a free museum with lots of children's activities (Horniman Gardens and Museum).
In the Greenwich area, which of course has one of the best parks in the world, you might want to consider Thomas Tallis for secondary. I'm not so familiar with the other schools there. Bexley, next borough out, has the 11+. The grammar schools are supposed to be very good but I don't know what the other schools are like. Both these areas are on a direct train line, depending on where you end up, into London Bridge which usually means stopping at New Cross or New Cross Gate.
Dulwich: expensive area but has very good state schools for all ages now. I think you'd probably be able to get a bus to New Cross - even if you have to take two buses it would be a fairly short journey - or else you could get the train to London Bridge and then it's one stop back out again (an expensive way to do it compared with the bus, cycling or walking). Another excellent park and there are the old woods to walk through.
Bromley: generally good schools across the borough, including one very high-powered girls' grammar school. You would be on a direct train line from Bromley South to Catford, where it is an easy switch to Catford Bridge station next door and three stops from there to New Cross.
Sutton: direct train line to New Cross Gate. Has several hugely oversubscribed grammar schools.
Good luck!