You've misunderstood the Wikipedia article, which is not surprising because it is poorly written. The government can establish mechanisms for people to provide statements "under penalty of perjury", such as IRS forms. But the mere words "under penalty of perjury" aren't a magic incantation; they have force only when specifically given it by the government.

"Under penalty of perjury" is just a mechanism for substituting a written declaration for an in-person swearing.

If you look at the statute, the governing condition is (paraphrased) "under laws or circumstances requiring or permitting a sworn statement".

IANAL, but 'anigbrowl has some training here.