I have the similar but earlier LWPro630, in which all ports (including simultaneous Appletalk over ethernet and localtalk) are indeed active at the same time. I haven't tried with a parallel connection, but the specs for the LWP630 say it will do that, too, i.e., all ports at once.

It should be the same for the LW16/600. These were professional quality printers, the 16/600 was a big step up from and the replacement for the LWPro630, and I can't see such features having been abandoned, especially as interoperability with DOS/ Windows was a major marketing theme of the time.

Does yours have the fax card?