Thanks Taoman,
Yes, parallel meaning the signal is distributed to all the attached devices together. It's the same as having a splitter wiring multiple phones together. The 'line jack' and 'phone jack' on the Bright ID'er box is just a convenient way to wire the box and a phone together without having to have a splitter. It's really not in series.
After I posted the last reply I had the same thought as far as trying with the Obi only driving the Bright ID'er, but it behaves the same - No Message. So it recognizes the ring voltage, but it's missing the caller ID tones.
This was working before I 'cut the cord'. And I have two of these Bright Id'ers and neither one of them are working, so I don't think it's the Bright ID'er.