How do I mount a sparsebundle disk image that was created on another machine attached to the network?

I think the DMG probably needs to be served by an OS X AFP service that can do conflict management for when multiple computers try to change the filesystem at the same time.

Since you indicated in the comments that each of the computers can mount the DMG individually, but not simultaneously, I bet that your NAS is detecting that the DMG file has been opened for writing by computer1 and is locking it from open+write by other computers until the first is done (unmounts it).

Ooooh, according to this Q/A, you can convert a DMG to read-only; then use a shadow file to keep track of changes. Just make sure to have a separate shadow file per computer and you should be set!

Please test this out with a sample DMG before converting your 12 TB DMG. Steps:

1. Create a trivial Read-Only DMG with a few files.
2. copy it to your NAS
3. In a terminal window hdiutil attach on /Volumes/NAS/path/to/your.dmg -shadow comp1.shadow
4. head over to another computer
5. run hdiutil attach on /Volumes/NAS/path/to/your.dmg -shadow comp2.shadow NOTE the different filename after -shadow, making sure that they don't both use the same shadow file.
6. see if it mounted