The sport is soccer. The round object you kick during your soccer game is a soccer ball. Note that even though the origin of the word soccer is British (it is an abbreviated version of "association football", with that "-er" ending found in such BE terms as "rugger" for rugby), it is usually thought of in Britain as an Americanism, as the sport is usually called "football" in the UK. As the term "football" in the US refers to a very different sport, another word had to be used -- and thus Americans imported the term "soccer".
You would say you played soccer at the park because soccer is the name of the game. You would not say you played soccer ball.
Since the actual object is a soccer ball, you can say you played with a soccer ball. This doesn't necessarily mean that you played soccer though. It could mean that you just tossed around the soccer ball.
With the word football (used to describe soccer in just about every other place in the world besides the USA ), you may say that you played football in the park.
You may also mean that you played with a football, but that would imply that you just tossed the ball around instead of playing an actual game of football.