CAUTION: NOT A TEACHER
(1) I think (repeat: think) that most Americans nowadays prefer "on."
(a) I went to the "books" section of Google and discovered that although some
Americans (especially in the earlier part of the twentieth century) do use "in," it
appears that "on" is now the preferred form for Americans.
(2) Here are just four examples:
(a) Swings are ... one of the most used pieces of equipment on a playground.
Source: Planning and Urban Design Standards (2006) by the American Planning Association.
(b) David and his friend, Brett, noticed two empty swings on the playground.
Source: One Year Book of Family Devotions (2000).
***
(c) A group of six children is actively playing on a permanent, modular playground system.
Source: Outdoor Play (2002) by Karyn Housen.
(d) These children also played together on the playground.
Source: Critical Perspectives on Early Childhood Education (1991) by Lois Weis.
(3) Why is "on" preferred? Maybe (maybe) because a playground is a flat area.
Therefore:
On a football field (as Rover said)
On a soccer field. (To Americans, "football" = American football, not soccer.)
On the tennis court
On the golf course
On the sidewalk
In his famous (and huge) grammar, Professor Quirk gives this example:
The players were practicing on the field . ( = surface for sports)
Cows were grazing in the field. ( = enclosed area of land)