Why in this sentence do we use を and not で。
Both sound okay to me and I don't think there's much difference in meaning... just the former sounds a bit more colloquial to me, I would write を if I was told to fill in the blank in 「公園( )散歩します。」 in Japanese class, but I think I usually say "公園で散歩してたらblah blah..." or "公園散歩してたらblah blah・・・" (leaving out the を/で) in daily conversation.
Maybe 公園で散歩します is 公園で散歩をします with its を dropped? You don't say 公園を散歩をします using を twice, so if you use 散歩をします then you can add 公園で but not 公園を.
I think you cannot say 公園を遊ぶ because the verb 遊ぶ cannot take the particle を.