I'm in the U.S. and considering printing t-shirts with foreign logos of national parks on them. The shirts would be sold online. In many cases, it's difficult to determine if these logos are trademarked. I can't see that these shirt would damage the parks in any way or compete with them. I'm also considering putting logos of various departments of foreign governments (i.e. Republic of South Africa's Arts and Culture Department). Is this use of any of these logos legal?
The logos are almost certainly under copyright - most countries in the world, unlike the US, do not put government work in the public domain.
They are also almost certainly trade marks: trade marks do not have to be registered.
You are eating into their market for sales of t-shirts so there is no fair use defence.
You can't do this.
There are actually two questions to ask here: "Is this a violation of Trademark" and "Is this a violation of Copyright".
The answers are "Probably" and "Almost Certainly".
Per the terms of Berne Convention, which is signed by most of the world, including South Africa, copyright attaches to a copyrightable work when "it is fixed in a tangible medium", like a t-shirt, or a piece of paper (which is your greater concern).
Not all logos are copyrightable, as the need to have "at least some level of creativity" involved. However, if you are putting them on a shirt, then they almost certainly have the request amount of creativity.
Bottom line: either don't do this, or first write to each foreign national park and request their permission to do so. Be aware that they are well within their rights to refuse, or to demand a fee or other compensation.