Would running a new subpanel in parallel meet NEC requirements? If not, what should I do?
A sub panel must have the neutral and ground isolated. Panels come with a very long, rather thick (about 1/4 x 20) green bonding screw that connects the neutral bar to the can in the case of a primary panel. You don't get a neutral from your utility, you create one with that bonding screw.

Sub panels should be fed with 3 insulated conductors of appropriate size, and a ground that need not be insulated (but can be, if you want). So the first part of your question is, yes, that sub panel must be grounded, but the grounding conductor should be attached to the can using a ground lug, not by landing it to the neutral bar.

Sub panels must also have a fused disconnect, which means they need to be fed from a breaker, and there can't be anything else on that breaker. Don't double tap. You have two options here:


Increase the size of the existing sub panel
Put in a new double pole breaker in the existing sub panel and use that to feed your new sub panel. Land the circuits you had to pull out of the existing sub panel into the new one.


As others have noted, you need to watch your loads. If all you need are a few convenience receptacles or a lighting circuit, you should probably be o.k. (hard to tell with what you've given).

Either way, get an amprobe and look at what each incoming phase is pulling in your existing sub panel prior to doing anything. Make sure everything is on when you do. If it's only pulling 25 - 30A on average, you should be o.k. to add a small 8 circuit sub panel. Since it's directly in the line of sight with the existing sub panel, the new sub panel need not have a main breaker since the means of disconnect is right next to it.

A six to eight circuit sub panel runs about $80 without breakers, they typically start at 50A, but you don't have to feed them with 50A. You could feed it from a 30A breaker if all you want are convenience receptacles and lights. 

Here is an amprobe being used:


(source: amprobe.com)  

Do that on your sub panel first (one phase at a time) just to be sure you have room to add more. If not, you need to replace your existing sub panel, and an electrician is really your best bet there.

Another good thing to do is measure the draw of the circuits you'll have to move to the sub panel in order to make room for the breaker that will feed it. Obviously, you want to move the circuits drawing the least to the new panel in the end. Some re-arranging might be needed to make that happen.

Since this is a garage, take care what you connect to the sub panel. If you are going to be powering something like a compressor (or anything else with a decent sized motor), carefully consider the locked rotor amperage when determining the load. It will be printed on the motor.

Finally, if any of this sounds overwhelming, call an electrician. If you get into any kind of trouble, call an electrician.