DISCLAIMER: I am not a lawyer, but I may have missed my calling

A discussion with +Be Chego has led me to discover a probable answer to "So, Who's Ordering All These Damn Nonsensical Closures, Anyway?"

The Answer to this question is (much to the dismay of plenty of folks who really like the image of a smug Barack Obama in the Oval Office with a list of 20,000+ programs on an old-timey scroll, ticking off Who's Naughty and Who's Nice boxes of what to close like some weird Santa Grinch) that it's NOT the White House.

It's very likely the Department of Justice who has made the call to close up shop, and that call would be based on an interpretation of existing law. Specifically:

It appears that all of the 'hinky' closures that have people scratching their collective heads are, in fact, required by the Antideficiency Act, The origin of which actually dates to the late 1800's. 
Wiki: http://en.wikipedia.org/wiki/Antideficiency_Act
Full Text:  http://www.law.cornell.edu/uscode/text/31/1341

In a nutshell – It seems that the unusual closures that we have not seen in the past are very likely a result of this being the first shutdown in a while to happen after Sequestration has already happened. 

Parks (and the contracts to operate them) are not on the list, so it would then be the Department of Justice that would most likely be the governmental unit who would consider the closure issue in light of what the law says, and issue the an opinion/memo indicating that the Federal Government is required to close anything not on the Sequestration Exception List.

Which means…Parks – even when the park is fully supported by user fees – are now required to be closed, even when in the past they have not – because the funds to pay for the people who run the (required) administration of the program that oversees fully user-supported parks have been sequestered since the Spring of 2013, and the Antideficiency Act³ specifically prohibits spending that money without prior authorization.

Therefore:
Since:
We have no bipartisan budget agreement, AND
We have Sequestration already in effect, AND
The National Park Service is not on the Sequestration Exemption List, AND
The National Park Service still provides mandated oversight of parks that are fully supported by user fees, AND
The National Park Service by law cannot currently provide that mandated oversight…

…even Parks that are fully supported by user fees are require to close.

TL;DR – User Funded Parks are closed because there's nobody at the National Park Service to oversee, or even take the NPS portion of the receipts from the Concessionaire-run park AND the NPS can't (currently) even commit to having someone there to take the reciepts or do the oversight in the future  – Therefore, the Concessionaire's park must close, even if the Concessionaire's park is largely self-sufficient, requires very little oversight, and hasn't seen a National Park Service employee physically on the premises in months or even years. 

SOURCES
¹
31 USC § 1341 – Limitations on expending and obligating amounts
(a)
(1) An officer or employee of the United States Government or of the District of Columbia government may not – 
(A) make or authorize an expenditure or obligation exceeding an amount available in an appropriation or fund for the expenditure or obligation;
(B) involve either government in a contract or obligation for the payment of money before an appropriation is made unless authorized by law;
(C) make or authorize an expenditure or obligation of funds required to be sequestered under section 252 of the Balanced Budget and Emergency Deficit Control Act of 1985; or
(D) involve either government in a contract or obligation for the payment of money required to be sequestered under section 252 of the Balanced Budget and Emergency Deficit Control Act of 1985 ² 
(2) This subsection does not apply to a corporation getting amounts to make loans (except paid in capital amounts) without legal liability of the United States Government.

² Section 252 of the Balanced Budget and Emergency Deficit Control Act of 1985:

Full Text: http://www.house.gov/legcoun/Comps/Balanced%20Budget%20And%20Emergency%20Deficit%20Control%20Act%20Of%201985–(Part%20C).pdf

Page 21 lists the exceptions to sequestration – i.e. "Stuff That We Can Still Legally Pay For".

³ which has existed in one form or another since 1870, when the military would blow their budget in a few months, then live off of coerced appropriation after coerced appropriation after the fact