Or, so I've always been told, and believed until I started reading the law.
Background: all entrances to buildings, as well as the driveways, are posted with signs warning that "It is a crime (felony) to bring any firearms, weapons, blah-blah-blah onto the grounds without the approval of the warden." The signs cite 18 USC 1791.
Sec. 1791. Providing or possessing contraband in prison
(a) Offense. - Whoever -
(1) in violation of a statute or a rule or order issued under a
statute, provides to an inmate of a prison a prohibited object,
or attempts to do so; or
(2) being an inmate of a prison, makes, possesses, or obtains,
or attempts to make or obtain, a prohibited object;
shall be punished as provided in subsection (b) of this section.
( . . . )
Note that the law only prohibits providing contraband to inmates, or possession of contraband by inmates. Nothing at all about possession by non-inmates.
I wondered about 1791(a)(1), and any rules or orders issued under the statute, and I found 28 CFR 6.1:
TITLE 28--JUDICIAL ADMINISTRATION
CHAPTER I--DEPARTMENT OF JUSTICE
PART 6--TRAFFIC IN CONTRABAND ARTICLES IN FEDERAL PENAL AND CORRECTIONAL INSTITUTIONS--Table of Contents
Sec. 6.1 Consent of warden or superintendent required.
The introduction or attempt to introduce into or upon the grounds of
any Federal penal or correctional institution or the taking or attempt
to take or send therefrom anything whatsoever without the knowledge and
consent of the warden or superintendent of such Federal penal or
correctional institution is prohibited.
That's the entire rule. A bit broad and vague, wot? The warden certainly doesn't know or give consent to the contents of my wallet or briefcase. (Should I turn myself in?

Since CFRs can only explain and enable implementation of the USC, not add to the law something it doesn't authorize, then any CFR based on 18 USC 1791 is restricted to providing contraband to inmates, or possession by inmates.
It's interesting that the BOP doesn't cite 18 USC 930, which most people are familiar with when it comes to the Post Office debate. Section 930 specifically defines "federal facilities" as "a building or part thereof owned or leased by the Federal Government, where Federal employees are regularly present for the purpose of performing their official duties".
Section 930 also requires that "Notice ... shall be posted conspicuously at each public entrance to each Federal facility ... and no person shall be convicted of an offense ... if such notice is not so posted ..."
So even under 18 USC 930, parking lots are not restricted.
I'm leaving out any discussion of workplace policies here, because that is a separate matter. But to the best of my ability to determine, the Bureau of Prisons position (that it is a felony to have a firearm anywhere on the grounds except with the warden's knowledge and consent), is not supported by the law.
I look forward to hearing thoughts from Charles, txinvestigator, and others.
Kevin