Hey Arod,
I would like to help you out, but you are going to have to give us more information. There isn't a serial read.vi. There are VISA Read.vi and Serial Port Read.vi, so I will need to know which one you are using. Also what version of VISA and LabVIEW are you using.
You should be able to run multiple VISA Read operations in parallel when they are using different VISA resources. This should work fine with LabVIEW 6.0.2 and higher, and NI-VISA 2.6 and higher. We have seen them work fine in parallel, as long as you ensure each VISA Read icon is set to be run asynchronously (you can right-click the icon and specify synchronous or asynchronous behavior). If they are synchronous, then that is the reason you are not seeing the parallel behavior you de
sire.
I hope this helps out.
-Josh