. . . I have always been confused on the law regarding "on post office property" and if it encompasses the parking lot or not. . . .
This is one of those situations where state law and Federal law may be in conflict. In Texas, nearly all parking lots are explicitly excluded from "premesis" restrictions on concealed carry. Federal law? Unclear. And in some cases, it's not at all clear that the USPS actually
owns the parking lot. Nobody opining on various websites dealing with Texas concealed carry has cited any case law that I've seen. (I'd welcome a link if case law exists!) Federal law mostly trumps state law, except where things like state's rights come into play, but it's not clear to me where that line exists, and I'm not interested in finding out the hard way.
There's also a school of legal thought - a
very small minority of legal thought - that says due to a supposed loophole in the verbiage of the Federal law, a Texas CHL holder
can carry concealed in a post office.
Nobody I know wants to be a test case, and so far I haven't seen any Texas case law establishing exactly what the limits are.
I've read and been told (in my original CHL class) that state/local LEOs in Texas have no interest in enforcing Federal law that conflicts with State law, but again, I don't want to be a test case.