No, no parking lots underground at Big Blue.
There never was and there never will be.
The underground tunnels are too narrow and the structure itself too old and tall to support further excavation. BB can explain, but the complex is not built to be earthquake-proof, as it is primarily concrete, and some areas, like Leb Hall, have structural damage. Excavating tunnels would never get building approval, there is a genuine risk of walls collapsing from the vibrations. I worked on-site for two years for the company that built the first 8 miles of subway track in LA, so I know a bit about this. There are also absolutely no entrances for a vehicle, the entrances are/were just stairwells or regular elevators. Even the few original freight elevators were never big enough for more than one or two golf cart type of vehicles.
The tunnels are about the width of 2-1/2 golf carts at best, but many are only wide enough for 2-3 people. There aren't any sub-tunnels beneath the tunnels, either. Not exactly. There is a short section beneath the main building that connects Leb Hall tunnels to main blg tunnels where it is kind of on two levels, though, and something similar where it connects to the old Pubs org on Sunset.
The only area wide enough for a car is the short tunnel between AOLA and ASHO that goes under the street. That tunnel is/was wide enough for two cars, but it was shut down and sectioned off and apart from ASHO some time in the early 80s for Code reasons. It had water and plumbing leaks then and some of the concrete was damaged. I suppose it's possible individuals might still use it, depending on how permanently they closed it, but there's nowhere for a car to go even if you could get one down there, as the tunnels become narrow again pretty quickly.
The Parking Structure is a separate, unconnected structure, its basement floor not fully underground and that's as far down as it goes, to the best of my memory, but I've only been in the lower level of it a couple of times. BB can confirm.
ADDED: Speaking of tunnels, there was chat a few years ago about a possible tunnel beneath the Hollywood Inn blg (the test center on Hollywood Blvd) and whatever Hubbard blgs are now across the street. The history of that section of Hollywood Blvd is that AFTER the company I worked for built our section of track in the late 80s, the contract was not renewed and LA Metra took over the building of track to save money. They decided to re-use the original tunnels under Hollywood Blvd. Our engineers disagreed. LA Metra went ahead and re-used the tunnels there anyway and built 2-3 miles of track directly under Hollywood Blvd. One night the tunnel collapsed, creating a huge buckle on the street. Nobody was in it at the time, so no one was hurt. Trying to cut costs and save millions cost LA billions. It was about a year before the street was repaired and the tunnel was shut down.
I doubt there is any way to enter that tunnel anymore - it is far too dangerous and would have been walled off by concrete and maybe even filled in completely, so that rumor was just hearsay.
History is documented here:
http://articles.latimes.com/2000/jun/18/news/mn-42276