From what we've seen of Beckenham, we love it!
Don't live there but we have viewed lots of properties over the past year, sadly it's out of our price range- we could only realistically afford the ex-council 2 bed flats which felt mega £££ for what you got.
Your budget is almost twice as much as ours so I'm sure you'll find a decent house for that amount.
There are lots of flats in Beckenham (appears to be a decent buy to let market, so many people looking for convenient commute into town) so I'd say that the parks will be busy as not everyone has their own garden.
We visited Kelsey (I think) park once and the playground area was rammed, plus lots of people in the park itself, although it was summer time.
I do agree with the schools thing- the primaries all seem great but there is nothing much at secondary level...think Sedgehill is the closest but it's Lewisham borough maybe?
Also, I noticed that a lot of vendors who owned 2 bed flats, they often had a young child and were looking to move to another part of Bromley borough to buy a house e.g. West Wickham, Hayes
So even with an extraordinary rise in price of their property (often the price had almost doubled in a few short years) they seemed priced out of the Beckenham market for an actual house.
Although, it could have been a secondary schools thing, but I don't know whether you look that far ahead with an 18 month old (we certainly haven't).
Good luck with the search!