a park, just because its public, does not make it public property. if it were public property, that would mean that anyone would be able to do whatever they wanted there. you'd be able to plant a garden anywhere (and others would be allowed to disrupt said garden). its still owned by someone, that someone being the government, which has granted it public use. so while those birds are non-native, if they are established there (meaning not just passing through, but actually living and breeding there), then the park does have the right to decide what happens to them.
if you had a hive of bees in your front yard, they would be yours to do what you saw fit. its different than a stray dog wandering onto your property.
and besides, the other reason to just ask is, you'll get a straight answer. they'll know if its legal or not. asking at the source is generally the best way to get information, especially since the park may have its own regulations, aside from federal, state, and county regulations. asking does no harm.