"[ball]park" in AmEng vernacular

Solution 1:

According to Wikipedia, the major league baseball venues are named as follows: 12 are "parks", such as PNC Park in Pittsburgh. 10 are "fields", such as Citi Field in New York. Only four are "stadiums," such as Yankee Stadium in New York. Then there is the O.co Coliseum in Oakland, and the Rogers Centre in Toronto. The "park" in Cincinnati actually contains "ball park" in its name: The Great American Ball Park. (Great American is an insurance sponsor, not a descriptor of the park.) And Atlanta's Turner Field will soon be replaced by SunTrust Park.

https://en.wikipedia.org/wiki/List_of_Major_League_Baseball_stadiums

However, no NFL football team plays in a "park." Nineteen play in "stadiums", nine in "fields", and two in "coliseums" (Oakland and now again Los Angeles). Two teams play in a "dome", one of which is a "superdome" (New Orleans).

https://en.wikipedia.org/wiki/List_of_current_National_Football_League_stadiums

As you can see, the titles of both articles include the word "stadiums", which is the all-purpose word for this type of building. But baseball is played in a park, and football in a stadium. Just ask George Carlin:

https://www.youtube.com/watch?v=qmXacL0Uny0

In Major League Soccer, there are 10 stadiums, two fields, one place, one center, one bowl, one arena and four parks. But many US soccer venues are owned by franchises in other sports, including one of the parks. None of the parks is called a "ball park." So you can draw your own conclusions.

Solution 2:

When referring to baseball, the word "baseball" is usually used in conjunction with the venue: baseball field, baseball stadium, baseball arena - unless the word "ballpark" is used, in which case, it is understood to be baseball.

"Ball field" is used to describe the playing ground for any sport where there is grass (or a reasonable facsimile) underfoot.